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GIMP

GNU Image Manipulation Program, commonly known by its acronym GIMP, is a free and open-source raster graphics editor used for image manipulation (retouching) and image editing, free-form drawing, transcoding between different image file formats, and more specialized tasks. It is extensible by means of plugins, and scriptable. It is not designed to be used for drawing, though some artists and creators have used it in this way.

GIMP is released under the GPL-3.0-or-later license and is available for Linux, macOS, and Microsoft Windows.

History
In 1995, Spencer Kimball and Peter Mattis began developing GIMP—originally named General Image Manipulation Program—as a semester-long project at the University of California, Berkeley for the eXperimental Computing Facility. The acronym was coined first, with the letter G being added to -IMP as a reference to "the gimp" in the scene from the 1994 film Pulp Fiction.

1996 was the initial public release of GIMP (0.54). The editor was quickly adopted and a community of contributors formed. The community began developing tutorials and artwork and sharing better work-flows and techniques.

In the following year, Kimball and Mattis met with Richard Stallman of the GNU Project while he visited UC Berkeley and asked if they could change General in the application's name to GNU (the name of the operating system created by Stallman), and Stallman approved. The application subsequently formed part of the GNU software collection.

The first release supported Unix systems, such as Linux, SGI IRIX and HP-UX. Since then, GIMP has been ported to other operating systems, including Microsoft Windows (1997, GIMP 1.1) and macOS.

A GUI toolkit called GTK (at the time known as the GIMP ToolKit) was developed to facilitate the development of GIMP. The development of the GIMP ToolKit has been attributed to Peter Mattis becoming disenchanted with the Motif toolkit GIMP originally used. Motif was used up until GIMP 0.60.

In recent versions (since the GIMP 2.9 build), the removal of the Lanczos image scaling algorithm, which had been used by GIMP and other image editing programs for many years, in favor of pushing forward the new NoHalo and LoHalo algorithms developed by Nicolas Robidoux, caused some controversy among GIMP users, with some users standing by the change but others expressing their dissatisfaction about it, due to mixed quality results in some image scaling scenarios, leading some users to keep using the older 2.8 version of GIMP simply because it's the last build with Lanczos support, and a few users giving up on using the application altogether as a result. To this day, several users hope to see a future version of GIMP with the Lanczos algorithm added back as an option for image resampling.

Mascot

GIMP's mascot is called Wilber and was created in GIMP by Tuomas Kuosmanen, known as tigert, on 25 September 1997. Wilber received additional accessories from other GIMP developers, which can be found in the Wilber Construction Kit, included in the GIMP source code as.

Development
GIMP is primarily developed by volunteers as a free and open source software project associated with both the GNU and GNOME projects. Development takes place in a public git source code repository, on public mailing lists and in public chat channels on the GIMPNET IRC network.

New features are held in public separate source code branches and merged into the main (or development) branch when the GIMP team is sure they won't damage existing functions. Sometimes this means that features that appear complete do not get merged or take months or years before they become available in GIMP.

GIMP itself is released as source code. After a source code release, installers and packages are made for different operating systems by parties who might not be in contact with the maintainers of GIMP.

The version number used in GIMP is expressed in a major-minor-micro format, with each number carrying a specific meaning: the first (major) number is incremented only for major developments (and is currently 2). The second (minor) number is incremented with each release of new features, with odd numbers reserved for in-progress development versions and even numbers assigned to stable releases; the third (micro) number is incremented before and after each release (resulting in even numbers for releases, and odd numbers for development snapshots) with any bug fixes subsequently applied and released for a stable version.

Previously, GIMP applied for several positions in the Google Summer of Code (GSoC). From 2006 to 2009 there have been nine GSoC projects that have been listed as successful, although not all successful projects have been merged into GIMP immediately. The healing brush and perspective clone tools and Ruby bindings were created as part of the 2006 GSoC and can be used in version 2.8.0 of GIMP, although there were three other projects that were completed and are later available in a stable version of GIMP; those projects being Vector Layers (end 2008 in 2.8 and master), and a JPEG 2000 plug-in (mid 2009 in 2.8 and master). Several of the GSoC projects were completed in 2008, but have been merged into a stable GIMP release later in 2009 to 2014 for Version 2.8.xx and 2.10.x. Some of them needed some more code work for the master tree.

Second public Development 2.9-Version was 2.9.4 with many deep improvements after initial Public Version 2.9.2. Third Public 2.9-Development version is Version 2.9.6. One of the new features is removing the 4 GB size limit of XCF file. Increase of possible threads to 64 is also an important point for modern parallel execution in actual AMD Ryzen and Intel Xeon processors. Version 2.9.8 included many bug fixes and improvements in gradients and clips. Improvements in performance and optimization beyond bug hunting were the development targets for 2.10.0. MacOS Beta is available with Version 2.10.4.

The next stable version in the roadmap is 3.0 with a GTK3 port. 2.99-Series is the development Series to 3.0. Jehan Pages, the lead developer and maintainer of GIMP, stated that GIMP 3.0's release is tentative for May 2024 and plans to announce the release at the next Libre Graphics Meeting conference.

GIMP developers meet during the annual Libre Graphics Meeting. Interaction designers from OpenUsability have also contributed to GIMP.

Distribution
The current version of GIMP works with numerous operating systems, including Linux, macOS and Windows. Many Linux distributions, such as Fedora Linux and Debian, include GIMP as a part of their desktop operating systems.

GIMP began to host its own downloads after discontinuing use of SourceForge in 2013. The website later repossessed GIMP's dormant account and hosted advertising-laden versions of GIMP for Windows.

In 2022, GIMP was published on the Microsoft Store for Windows.

Professional reviews
Lifewire reviewed GIMP favorably in March 2019, writing that "[f]or those who have never experienced Photoshop, GIMP is simply a very powerful image manipulation program," and "[i]f you're willing to invest some time learning it, it can be a very good graphics tool."

GIMP's fitness for use in professional environments is regularly reviewed; it is often compared to and suggested as a possible replacement for Adobe Photoshop.

GIMP 2.6 was used to create nearly all of the art in Lucas the Game, an independent video game by developer Timothy Courtney. Courtney started development of Lucas the Game in early 2014, and the video game was published in July 2015 for PC and Mac. Courtney explains GIMP is a powerful tool, fully capable of large professional projects, such as video games.

The single-window mode introduced in GIMP 2.8 was reviewed in 2012 by Ryan Paul of Ars Technica, who noted that it made the user experience feel "more streamlined and less cluttered". Michael Burns, writing for Macworld in 2014, described the single-window interface of GIMP 2.8.10 as a "big improvement".

In his review of GIMP for ExtremeTech in October 2013, David Cardinal noted that GIMP's reputation of being hard to use and lacking features has "changed dramatically over the last couple years", and that it was "no longer a crippled alternative to Photoshop". He described GIMP's scripting as one of its strengths, but also remarked that some of Photoshop's features – such as Text, 3D commands, Adjustment Layers and History –  are either less powerful or missing in GIMP. Cardinal favorably described the UFRaw converter for raw images used with GIMP, noting that it still "requires some patience to figure out how to use those more advanced capabilities". Cardinal stated that GIMP is "easy enough to try" despite not having as well developed documentation and help system as those for Photoshop, concluding that it "has become a worthy alternative to Photoshop for anyone on a budget who doesn't need all of Photoshop's vast feature set".

The user interface has been criticized for being "hard to use".

Features

Tools used to perform image editing can be accessed via the toolbox, through menus and dialogue windows. They include filters and brushes, as well as transformation, selection, layer and masking tools. GIMP's developers have asserted that it has, or at least aspire to it having, similar functionality to Photoshop, but has a different user interface. Also, as of 2024 and version 2.10, a fundamental and essential difference between GIMP, on one hand, and major commercial software like Photoshop and Serif Affinity Photo, on the other, is that very few of GIMP's editing operations occur as non-destructive edits, unlike the main commercial software.

Color
There are several ways of selecting colors, including palettes, color choosers and using an eyedropper tool to select a color on the canvas. The built-in color choosers include RGB/HSV/LAB/LCH selector or scales, water-color selector, CMYK selector and a color-wheel selector. Colors can also be selected using hexadecimal color codes, as used in HTML color selection. GIMP has native support for indexed color and RGB color spaces; other color spaces are supported using decomposition, where each channel of the new color space becomes a black-and-white image. CMYK, LAB and HSV (hue, saturation, value) are supported this way. Color blending can be achieved using the Blend tool, by applying a gradient to the surface of an image and using GIMP's color modes. Gradients are also integrated into tools such as the brush tool, when the user paints this way the output color slowly changes. There are a number of default gradients included with GIMP; a user can also create custom gradients with tools provided. Gradient plug-ins are also available.

Selections and paths
GIMP selection tools include a rectangular and circular selection tool, free select tool, and fuzzy select tool (also known as magic wand). More advanced selection tools include the select by color tool for selecting contiguous regions of color—and the scissors select tool, which creates selections semi-automatically between areas of highly contrasting colors. GIMP also supports a quick mask mode where a user can use a brush to paint the area of a selection. Visibly this looks like a red colored overlay being added or removed. The foreground select tool is an implementation of Simple interactive object extraction (SIOX), a method used to perform the extraction of foreground elements, such as a person or a tree in focus. The Paths Tool allows a user to create vectors (also known as Bézier curves). Users can use paths to create complex selections, including around natural curves. They can paint (or "stroke") the paths with brushes, patterns, or various line styles. Users can name and save paths for reuse.

Image editing
There are many tools that can be used for editing images in GIMP. The more common tools include a paint brush, pencil, airbrush, eraser and ink tools used to create new or blended pixels. The Bucket Fill tool can be used to fill a selection with a color or pattern. The Blend tool can be used to fill a selection with a color gradient. These color transitions can be applied to large regions or smaller custom path selections.

GIMP also provides "smart" tools that use a more complex algorithm to do things that otherwise would be time-consuming or impossible. These include:
 * Clone tool, which copies pixels using a brush
 * Healing brush, which copies pixels from an area and corrects tone and color
 * Perspective clone tool, which works like the clone tool but corrects for distance changes
 * Blur and sharpen tools
 * The Smudge tool can be used to subtly smear a selection where it stands
 * Dodge and burn tool is a brush that makes target pixels lighter (dodges) or darker (burns)

Layers, layer masks and channels
An image being edited in GIMP can consist of many layers in a stack. The user manual suggests that "A good way to visualize a GIMP image is as a stack of transparencies," where in GIMP terminology, each level (analogous to a transparency) is called a layer. Each layer in an image is made up of several channels. In an RGB image, there are normally 3 or 4 channels, each consisting of a red, green and blue channel. Color sublayers look like slightly different gray images, but when put together they make a complete image. The fourth channel that may be part of a layer is the alpha channel (or layer mask). This channel measures opacity where a whole or part of an image can be completely visible, partially visible or invisible. Each layer has a layer mode that can be set to change the colors in the image.

Text layers can be created using the text tool, allowing a user to write on an image. Text layers can be transformed in several ways, such as converting them to a path or selection.

Automation, scripts and plug-ins
GIMP has approximately 150 standard effects and filters, including Drop Shadow, Blur, Motion Blur and Noise.

GIMP operations can be automated with scripting languages. The Script-Fu is a Scheme-based language implemented using a TinyScheme interpreter built into GIMP. GIMP can also be scripted in Perl, Python (Python-Fu),  or Tcl, using interpreters external to GIMP. New features can be added to GIMP not only by changing program code (GIMP core), but also by creating plug-ins. These are external programs that are executed and controlled by the main GIMP program. MathMap is an example of a plug-in written in C.

There is support for several methods of sharpening and blurring images, including the blur and sharpen tool. The unsharp mask tool is used to sharpen an image selectively – it sharpens only those areas of an image that are sufficiently detailed. The Unsharp Mask tool is considered to give more targeted results for photographs than a normal sharpening filter. The Selective Gaussian Blur tool works in a similar way, except it blurs areas of an image with little detail.

GIMP-ML is an extension for machine learning with 15 filters.

GEGL
The Generic Graphics Library (GEGL) was first introduced as part of GIMP on the 2.6 release of GIMP. This initial introduction does not yet exploit all of the capabilities of GEGL; as of the 2.6 release, GIMP can use GEGL to perform high bit-depth color operations; because of this, less information is lost when performing color operations. When GEGL is fully integrated, GIMP will have a higher color bit depth and better non-destructive work-flow. GIMP 2.8.xx supports only 8-bit color, which is much lower than digital cameras, e.g., produce (12-bit or higher). Full support for high bit depth is included with GIMP 2.10. OpenCL enables hardware acceleration for some operations.

CTX
CTX is a new rasterizer for vector graphics in GIMP 3.0. Some simple objects, like lines and circles, can be reduced to vector objects.

File formats
GIMP supports importing and exporting with a large number of different file formats. GIMP's native format XCF is designed to store all information GIMP can contain about an image; XCF is named after the eXperimental Computing Facility where GIMP was authored. Import and export capability can be extended to additional file formats by means of plug-ins. XCF file size is extended to more than 4 GB since 2.9.6 and new stable tree 2.10.x.

Forks and derivatives
Because of the free and open-source nature of GIMP, several forks, variants and derivatives of the computer program have been created to fit the needs of their creators. While GIMP is cross-platform, variants of GIMP may not be. These variants are neither hosted nor linked on the GIMP site. The GIMP site does not host GIMP builds for Windows or Unix-like operating systems either, although it does include a link to a Windows build.

Forks

 * CinePaint: Formerly Film Gimp, it is a fork of GIMP version 1.0.4, used for frame-by-frame retouching of feature film. CinePaint supports up to 32-bit IEEE-floating point color depth per channel, as well as color management and HDR. CinePaint is used primarily within the film industry due mainly to its support of high-fidelity image formats. It is available for BSD, Linux, and macOS.
 * GIMP classic: A patch against GIMP v2.6.8 source code created to undo changes made to the user interface in GIMP v2.4 through v2.6. A build of GIMP classic for Ubuntu is available. As of March 2011, a new patch could be downloaded that patches against the experimental GIMP v2.7.
 * GIMP Portable: A portable version of GIMP for Microsoft Windows XP or later that preserves brushes and presets between computers.
 * GIMPshop: Derivative that aimed to replicate the Adobe Photoshop in some form. Development of GIMPshop was halted in 2006 and the project disavowed by the developer, Scott Moschella, after an unrelated party registered "GIMPshop" as part of an Internet domain name and passed off the website as belonging to Moschella while accepting donations and making revenue from advertising but passing on none of the income to Moschella.
 * GimPhoto: GimPhoto follows the Photoshop-UI tradition of GIMPshop. More modifications are possible with the GimPad tool. GimPhoto stands at version 24.1 for Linux and Windows (based on GIMP v2.4.3) and version 26.1 on macOS (based on GIMP v2.6.8). Installers are included for Windows 7, 8.1, and 10; macOS 10.6+; Ubuntu 14 and Fedora; as well as source code. Only one developer is at work in this project, so fast updates and new versions based on Gimp 2.8.x or 2.9.x are not planned.
 * McGimp: An independent port for macOS that is aim to run GIMP directly on this platform, and integrated multiple plug-ins intended to optimize photos.
 * Seashore: easier to use image editing application for macOS.
 * Glimpse: a discontinued fork of GIMP that was started because the word "gimp" is also a derogatory word for disabled people.

Extensions

GIMP's functionality can be extended with plugins. Notable ones include:
 * GIMP-ML, which provides machine learning-based image enhancement. GIMP-ML with python 3 is next target in development.
 * GIMP Animation Package (GAP), official plugin for creating animations. GAP can save animations in several formats, including GIF and AVI.
 * Resynthesizer, which provides content-aware fill. Original part of Paul Harrison's PhD thesis, now maintained by Lloyd Konneker.
 * G'MIC, which adds image filters and effects.

About GIMP

 * Libre Graphics Meeting
 * List of computing mascots
 * Category:Computing mascots

Other

 * Comparison of raster graphics editors
 * List of raster graphics editors
 * List of free and open-source software packages
 * List of 2D graphics software

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Global illumination

Global illumination (GI), or indirect illumination, is a group of algorithms used in 3D computer graphics that are meant to add more realistic lighting to 3D scenes. Such algorithms take into account not only the light that comes directly from a light source (direct illumination), but also subsequent cases in which light rays from the same source are reflected by other surfaces in the scene, whether reflective or not (indirect illumination).

Theoretically, reflections, refractions, and shadows are all examples of global illumination, because when simulating them, one object affects the rendering of another (as opposed to an object being affected only by a direct source of light). In practice, however, only the simulation of diffuse inter-reflection or caustics is called global illumination.

Algorithms
Images rendered using global illumination algorithms often appear more photorealistic than those using only direct illumination algorithms. However, such images are computationally more expensive and consequently much slower to generate. One common approach is to compute the global illumination of a scene and store that information with the geometry (e.g., radiosity). The stored data can then be used to generate images from different viewpoints for generating walkthroughs of a scene without having to go through expensive lighting calculations repeatedly.

Radiosity, ray tracing, beam tracing, cone tracing, path tracing, volumetric path tracing, Metropolis light transport, ambient occlusion, photon mapping, signed distance field and image-based lighting are all examples of algorithms used in global illumination, some of which may be used together to yield results that are not fast, but accurate.

These algorithms model diffuse inter-reflection which is a very important part of global illumination; however most of these (excluding radiosity) also model specular reflection, which makes them more accurate algorithms to solve the lighting equation and provide a more realistically illuminated scene. The algorithms used to calculate the distribution of light energy between surfaces of a scene are closely related to heat transfer simulations performed using finite-element methods in engineering design.

Photorealism
Achieving accurate computation of global illumination in real-time remains difficult. In real-time 3D graphics, the diffuse inter-reflection component of global illumination is sometimes approximated by an "ambient" term in the lighting equation, which is also called "ambient lighting" or "ambient color" in 3D software packages. Though this method of approximation (also known as a "cheat" because it's not really a global illumination method) is easy to perform computationally, when used alone it does not provide an adequately realistic effect. Ambient lighting is known to "flatten" shadows in 3D scenes, making the overall visual effect more bland. However, used properly, ambient lighting can be an efficient way to make up for a lack of processing power.

Procedure
More and more specialized algorithms are used in 3D programs that can effectively simulate the global illumination. These algorithms are numerical approximations of the rendering equation. Well known algorithms for computing global illumination include path tracing, photon mapping and radiosity. The following approaches can be distinguished here: In Light-path notation global lighting the paths of the type L (D | S) corresponds * E.
 * Inversion: $$L = (1-T)^{-1} L^e\,$$
 * is not applied in practice
 * Expansion: $$L = \sum_{i=0}^\infty T^iL^e$$
 * bi-directional approach: Photon mapping + Distributed ray tracing, Bi-directional path tracing, Metropolis light transport
 * Iteration: $$L_n tl_ e + = L ^{(n-1)}$$
 * Radiosity

A full treatment can be found in

Image-based lighting
Another way to simulate real global illumination is the use of high-dynamic-range images (HDRIs), also known as environment maps, which encircle and illuminate the scene. This process is known as image-based lighting.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Global illumination


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Geometric series

In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series

 * $$\frac{1}{2} \,+\, \frac{1}{4} \,+\, \frac{1}{8} \,+\, \frac{1}{16} \,+\, \cdots$$

is geometric, because each successive term can be obtained by multiplying the previous term by $$1/2$$. In general, a geometric series is written as $$a + ar + ar^2 + ar^3 + ...$$, where $$a$$ is the coefficient of each term and $$r$$ is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the Fourier series, and the matrix exponential.

The name geometric series indicates each term is the geometric mean of its two neighboring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighboring terms.

Coefficient a
The geometric series a + ar + ar2 + ar3 + ... is written in expanded form. Every coefficient in the geometric series is the same. In contrast, the power series written as a0 + a1r + a2r2 + a3r3 + ... in expanded form has coefficients ai that can vary from term to term. In other words, the geometric series is a special case of the power series. The first term of a geometric series in expanded form is the coefficient a of that geometric series.

In addition to the expanded form of the geometric series, there is a generator form of the geometric series written as

 * $$\sum^{\infty}_{k=0} a r^k$$

and a closed form of the geometric series written as

 * $$\frac{a}{1-r} \text{ for } |r|<1. $$

The derivation of the closed form from the expanded form is shown in this article's section. However even without that derivation, the result can be confirmed with long division: a divided by (1 - r) results in a + ar + ar2 + ar3 + ..., which is the expanded form of the geometric series.

It is often a convenience in notation to set the series equal to the sum s and work with the geometric series

 * s = a + ar + ar2 + ar3 + ar4 + ... in its normalized form
 * s / a = 1 + r + r2 + r3 + r4 + ... or in its normalized vector form
 * s / a = [1 1 1 1 1 ...][1 r r2 r3 r4 ...]T or in its normalized partial series form
 * sn / a = 1 + r + r2 + r3 + r4 + ... + rn, where n is the power (or degree) of the last term included in the partial sum sn.

Changing even one of the coefficients to something other than coefficient a would change the resulting sum of functions to some function other than a / (1 − r) within the range |r| < 1. As an aside, a particularly useful change to the coefficients is defined by the Taylor series, which describes how to change the coefficients so that the sum of functions converges to any user selected, sufficiently smooth function within a range.

Common ratio r

The geometric series a + ar + ar2 + ar3 + ... is an infinite series defined by just two parameters: coefficient a and common ratio r. Common ratio r is the ratio of any term with the previous term in the series. Or equivalently, common ratio r is the term multiplier used to calculate the next term in the series. The following table shows several geometric series:

The convergence of the geometric series depends on the value of the common ratio r:
 * If |r| < 1, the terms of the series approach zero in the limit (becoming smaller and smaller in magnitude), and the series converges to the sum a / (1 - r).
 * If |r| = 1, the series does not converge. When r = 1, all of the terms of the series are the same and the series is infinite. When r = −1, the terms take two values alternately (for example, 2, −2, 2, −2, 2,... ). The sum of the terms oscillates between two values (for example, 2, 0, 2, 0, 2,... ). This is a different type of divergence. See for example Grandi's series: 1 − 1 + 1 − 1 + ···.
 * If |r| > 1, the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, and the series does not converge to a sum. (The series diverges.)

The rate of convergence also depends on the value of the common ratio r. Specifically, the rate of convergence gets slower as r approaches 1 or −1. For example, the geometric series with a = 1 is 1 + r + r2 + r3 + ... and converges to 1 / (1 - r) when |r| < 1. However, the number of terms needed to converge approaches infinity as r approaches 1 because a / (1 - r) approaches infinity and each term of the series is less than or equal to one. In contrast, as r approaches −1 the sum of the first several terms of the geometric series starts to converge to 1/2 but slightly flips up or down depending on whether the most recently added term has a power of r that is even or odd. That flipping behavior near r = −1 is illustrated in the adjacent image showing the first 11 terms of the geometric series with a = 1 and |r| < 1. The common ratio r and the coefficient a also define the geometric progression, which is a list of the terms of the geometric series but without the additions. Therefore the geometric series a + ar + ar2 + ar3 + ... has the geometric progression (also called the geometric sequence) a, ar, ar2, ar3, ... The geometric progression - as simple as it is - models a surprising number of natural phenomena,
 * from some of the largest observations such as the expansion of the universe where the common ratio r is defined by Hubble's constant,
 * to some of the smallest observations such as the decay of radioactive carbon-14 atoms where the common ratio r is defined by the half-life of carbon-14.

As an aside, the common ratio r can be a complex number such as |r|eiθ where |r| is the vector's magnitude (or length), θ is the vector's angle (or orientation) in the complex plane and i2 = -1. With a common ratio |r|eiθ, the expanded form of the geometric series is a + a|r|eiθ + a|r|2ei2θ + a|r|3ei3θ + ... Modeling the angle θ as linearly increasing over time at the rate of some angular frequency ω0 (in other words, making the substitution θ = ω0t), the expanded form of the geometric series becomes a + a|r|eiω0t + a|r|2ei2ω0t + a|r|3ei3ω0t + ..., where the first term is a vector of length a not rotating at all, and all the other terms are vectors of different lengths rotating at harmonics of the fundamental angular frequency ω0. The constraint |r|<1 is enough to coordinate this infinite number of vectors of different lengths all rotating at different speeds into tracing a circle, as shown in the adjacent video. Similar to how the Taylor series describes how to change the coefficients so the series converges to a user selected sufficiently smooth function within a range, the Fourier series describes how to change the coefficients (which can also be complex numbers in order to specify the initial angles of vectors) so the series converges to a user selected periodic function.

Sum

The sum of the first n terms of a geometric series, up to and including the r n-1 term, is given by the closed-form formula:

$$ \begin{align} s_n &= ar^0 + ar^1 + \cdots + ar^{n-1}\\ &= \sum_{k=0}^{n-1} ar^k = \sum_{k=1}^{n} ar^{k-1}\\ &= \begin{cases} an                               & r = 1\\ a\left(\frac{1-r^{n}}{1-r}\right) & \text{otherwise} \end{cases} \end{align} $$

where $r$ is the common ratio. One can derive that closed-form formula for the partial sum, sn, by subtracting out the many self-similar terms as follows: $$ \begin{align} s_n &= ar^0 + ar^1 + \cdots + ar^{n-1},\\ rs_n &= ar^1 + ar^2 + \cdots + ar^{n},\\ s_n - rs_n &= ar^0 - ar^{n},\\ s_n\left(1-r\right) &= a\left(1-r^{n}\right),\\ s_n &= a\left(\frac{1-r^{n}}{1-r}\right), \text{ for } r \neq 1. \end{align} $$

As $n$ approaches infinity, the absolute value of $r$ must be less than one for the series to converge. The sum then becomes

$$ \begin{align} s &= a+ar+ar^2+ar^3+ar^4+\cdots\\ &= \sum_{k=0}^\infty ar^{k} = \sum_{k=1}^\infty ar^{k-1}\\ &= \frac{a}{1-r}, \text{ for } |r|<1. \end{align} $$

The formula also holds for complex $r$, with the corresponding restriction that the modulus of $r$ is strictly less than one.

As an aside, the question of whether an infinite series converges is fundamentally a question about the distance between two values: given enough terms, does the value of the partial sum get arbitrarily close to the finite value it is approaching? In the above derivation of the closed form of the geometric series, the interpretation of the distance between two values is the distance between their locations on the number line. That is the most common interpretation of the distance between two values. However, the p-adic metric, which has become a critical notion in modern number theory, offers a definition of distance such that the geometric series 1 + 2 + 4 + 8 + ... with a = 1 and r = 2 actually does converge to a / (1 - r) = 1 / (1 - 2) = -1 even though r is outside the typical convergence range |r| < 1.

Proof of convergence
We can prove that the geometric series converges using the sum formula for a geometric progression: $$\begin{align} 1 + r + r^2 + r^3 + \cdots \ &= \lim_{n\to\infty} \left(1 + r + r^2 + \cdots + r^n\right) \\ &= \lim_{n\to\infty} \frac{1-r^{n+1}}{1-r}. \end{align}$$ The second equality is true because if $$ |r| < 1,$$ then $$r^{n+1} \to 0$$ as $$n \to \infty$$ and $$ \begin{align} \left(1 + r + r^2 + \cdots + r^n\right)(1 - r) &= \left((1-r) + (r - r^2) + (r^2 - r^3) + \dots + (r^n - r^{n+1})\right) \\ &= 1 + (-r + r) + ( -r^2 + r^2) + \dots + (-r^n + r^n) - r^{n+1} \\ &= 1-r^{n+1}. \end{align} $$

Alternatively, a geometric interpretation of the convergence is shown in the adjacent diagram. The area of the white triangle is the series remainder = s − sn = arn+1 / (1 − r). Each additional term in the partial series reduces the area of that white triangle remainder by the area of the trapezoid representing the added term. The trapezoid areas (i.e., the values of the terms) get progressively thinner and shorter and closer to the origin. In the limit, as the number of trapezoids approaches infinity, the white triangle remainder vanishes as it is filled by trapezoids and therefore sn converges to s, provided |r|<1. In contrast, if |r|>1, the trapezoid areas representing the terms of the series instead get progressively wider and taller and farther from the origin, not converging to the origin and not converging as a series.

Rate of convergence

After knowing that a series converges, there are some applications in which it is also important to know how quickly the series converges. For the geometric series, one convenient measure of the convergence rate is how much the previous series remainder decreases due to the last term of the partial series. Given that the last term is arn and the previous series remainder is s - sn-1 = arn / (1 - r)), this measure of the convergence rate of the geometric series is arn / (arn / (1 - r)) = 1 - r, if 0 ≤ r < 1.

If r < 0, adjacent terms in the geometric series alternate between being positive and negative. A geometric interpretation of a converging alternating geometric series is shown in the adjacent diagram in which the areas of the negative terms are shown below the x axis. Pairing and summing each positive area with its negative smaller area neighbor results in non-overlapped trapezoids separated by gaps. To remove the gaps, broaden each trapezoid to cover the rightmost 1 - r2 of the original triangle area instead of just the rightmost 1 - |r|. However, to maintain the same trapezoid areas during this broadening transformation, scaling is needed: scale*(1 - r2) = (1 - |r|), or scale = (1 - |r|) / (1 - r2) = (1 + r) / (1 - r2) = (1 + r) / ((1 + r)(1 - r)) = 1 / (1 - r) where -1 < r ≤ 0. Note that because r < 0 this scale decreases the amplitude of the separated trapezoids in order to fill in the separation gaps. In contrast, for the case r > 0 the same scale 1 / (1 - r) increases the amplitude of the non-overlapped trapezoids in order to account for the loss of the overlapped areas.

With the gaps removed, pairs of terms in a converging alternating geometric series become a converging (non-alternating) geometric series with common ratio r2 to account for the pairing of terms, coefficient a = 1 / (1 - r) to account for the gap filling, and the degree (i.e., highest powered term) of the partial series called m instead of n to emphasize that terms have been paired. Similar to the r > 0 case, the r < 0 convergence rate = ar2m / (s - sm-1) = 1 - r2, which is the same as the convergence rate of a non-alternating geometric series if its terms were similarly paired. Therefore, the convergence rate does not depend upon n or m and, perhaps more surprising, does not depend upon the sign of the common ratio. One perspective that helps explain the variable rate of convergence that is symmetric about r = 0 is that each added term of the partial series makes a finite contribution to the infinite sum at r = 1 and each added term of the partial series makes a finite contribution to the infinite slope at r = -1.

Finite series
To derive this formula, first write a general geometric series as: $$\sum_{k=1}^{n} ar^{k-1} = ar^0 + ar^1 + ar^2 + ar^3 + \cdots + ar^{n-1}. $$

We can find a simpler formula for this sum by multiplying both sides of the above equation by 1 − r, and we'll see that

$$\begin{align} (1-r) \sum_{k=1}^{n} ar^{k-1} & = (1-r)\left(ar^0 + ar^1+ar^2+ar^3+\cdots+ar^{n-1}\right) \\ & = ar^0 + ar^1+ar^2+ar^3+\cdots+ar^{n-1} - ar^1-ar^2-ar^3-\cdots-ar^{n-1} - ar^n \\ & = a - ar^n \end{align}$$

since all the other terms cancel. If r ≠ 1, we can rearrange the above to get the convenient formula for a geometric series that computes the sum of n terms:

$$\sum_{k=1}^{n} ar^{k-1} = \frac{a(1-r^n)}{1-r}.$$

 * Related formulas

If one were to begin the sum not from k=1 or 0 but from a different value, say $a⁄1 − r$, then

$$\begin{align} \sum_{k=m}^n ar^{k-1} &= \begin{cases} \frac{a(r^{m-1}-r^n)}{1-r} & \text{if }r \neq 1 \\ a(n-m+1) & \text{if }r = 1 \end{cases}\\ \sum_{k=m}^n ar^k &= \begin{cases} a(n-m+1) & \text{if }r = 1 \\ \frac{a(r^m-r^{n+1})}{1-r} & \text{if }r \neq 1 \end{cases}\end{align}$$

Differentiating this formula with respect to $m$ allows us to arrive at formulae for sums of the form

$$G_s(n, r) := \sum_{k=0}^n k^s r^k.$$

For example:

$$\frac{d}{dr} \sum_{k=0}^nr^k = \sum_{k=1}^n kr^{k-1}= \frac{1-r^{n+1}}{(1-r)^2}-\frac{(n+1)r^n}{1-r}.$$

For a geometric series containing only even powers of $r$ multiply by $r$:

$$\begin{align} (1-r^2) \sum_{k=0}^{n} ar^{2k} &= a-ar^{2n+2}\\ \sum_{k=0}^{n} ar^{2k} &= \frac{a(1-r^{2n+2})}{1-r^2} \end{align}$$

Equivalently, take $1-r^2$ as the common ratio and use the standard formulation.

For a series with only odd powers of $r^2$, $$\begin{align} (1-r^2) \sum_{k=0}^{n} ar^{2k+1} &= ar-ar^{2n+3}\\ \sum_{k=0}^{n} ar^{2k+1} &= \frac{ar(1-r^{2n+2})}{1-r^2} &= \frac{a(r-r^{2n+3})}{1-r^2} \end{align}$$

An exact formula for the generalized sum $$G_s(n, r)$$ when $$s \in \mathbb{N}$$ is expanded by the Stirling numbers of the second kind as

$$G_s(n, r) = \sum_{j=0}^s \left\lbrace{s \atop j}\right\rbrace x^j \frac{d^j}{dx^j}\left[\frac{1-x^{n+1}}{1-x}\right].$$

Infinite series
An infinite geometric series is an infinite series whose successive terms have a common ratio. Such a series converges if and only if the absolute value of the common ratio is less than one ($r$ < 1). Its value can then be computed from the finite sum formula $$\sum_{k=0}^\infty ar^k = \lim_{n\to\infty}{\sum_{k=0}^{n} ar^k} = \lim_{n\to\infty}\frac{a(1-r^{n+1})}{1-r}= \frac{a}{1-r} - \lim_{n\to\infty}{\frac{ar^{n+1}}{1-r}} $$

Since: $$ r^{n+1} \to 0 \mbox{ as } n \to \infty \mbox{ when } |r| < 1.$$

Then: $$\sum_{k=0}^\infty ar^k = \frac{a}{1-r} - 0 = \frac{a}{1-r}$$

For a series containing only even powers of $$r$$, $$\sum_{k=0}^\infty ar^{2k} = \frac{a}{1-r^2}$$ and for odd powers only, $$\sum_{k=0}^\infty ar^{2k+1} = \frac{ar}{1-r^2}$$

In cases where the sum does not start at k = 0, $$\sum_{k=m}^\infty ar^k=\frac{ar^m}{1-r}$$ The formulae given above are valid only for $|r|$ < 1. The latter formula is valid in every Banach algebra, as long as the norm of r is less than one, and also in the field of p-adic numbers if $|r|$p < 1. As in the case for a finite sum, we can differentiate to calculate formulae for related sums. For example, $$\frac{d}{dr} \sum_{k=0}^\infty r^k = \sum_{k=1}^\infty kr^{k-1} = \frac{1}{(1-r)^2}$$

This formula only works for $|r|$ < 1 as well. From this, it follows that, for $|r|$ < 1, $$\sum_{k=0}^{\infty} k r^k = \frac{r}{\left(1-r\right)^2} \,;\, \sum_{k=0}^{\infty} k^2 r^k = \frac{r \left( 1+r \right)}{\left(1-r\right)^3} \, ; \, \sum_{k=0}^{\infty} k^3 r^k = \frac{r \left( 1+4 r + r^2\right)}{\left( 1-r\right)^4}$$

Also, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ⋯ is an elementary example of a series that converges absolutely.

It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is $$\frac12+\frac14+\frac18+\frac{1}{16}+\cdots=\frac{1/2}{1-(+1/2)} = 1.$$

The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely.

It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is $$\frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)} = \frac13.$$

Complex series
The summation formula for geometric series remains valid even when the common ratio is a complex number. In this case the condition that the absolute value of r be less than 1 becomes that the modulus of r be less than 1. It is possible to calculate the sums of some non-obvious geometric series. For example, consider the proposition $$ \sum_{k=0}^{\infty} \frac{\sin(kx)}{r^k} = \frac{r \sin(x)}{1 + r^2 - 2 r \cos(x)} $$

The proof of this comes from the fact that $$\sin(kx) = \frac{e^{ikx} - e^{-ikx}}{2i}, $$ which is a consequence of Euler's formula. Substituting this into the original series gives $$ \sum_{k=0}^{\infty} \frac{\sin(kx)}{r^k} = \frac{1}{2 i} \left[ \sum_{k=0}^{\infty} \left( \frac{e^{ix}}{r} \right)^k - \sum_{k=0}^{\infty} \left(\frac{e^{-ix}}{r}\right)^k\right].$$

This is the difference of two geometric series, and so it is a straightforward application of the formula for infinite geometric series that completes the proof.

Zeno of Elea (c.495 – c.430 BC)
2,500 years ago, Greek mathematicians had a problem when walking from one place to another: they thought that an infinitely long list of numbers greater than zero summed to infinity. Therefore, it was a paradox when Zeno of Elea pointed out that in order to walk from one place to another, you first have to walk half the distance, and then you have to walk half the remaining distance, and then you have to walk half of that remaining distance, and you continue halving the remaining distances an infinite number of times because no matter how small the remaining distance is you still have to walk the first half of it. Thus, Zeno of Elea transformed a short distance into an infinitely long list of halved remaining distances, all of which are greater than zero. And that was the problem: how can a distance be short when measured directly and also infinite when summed over its infinite list of halved remainders? The paradox revealed something was wrong with the assumption that an infinitely long list of numbers greater than zero summed to infinity.

Euclid of Alexandria (c.300 BC)

Euclid's Elements of Geometry Book IX, Proposition 35, proof (of the proposition in adjacent diagram's caption):

"Let AA', BC, DD', EF be any multitude whatsoever of continuously proportional numbers, beginning from the least AA'. And let BG and FH, each equal to AA', have been subtracted from BC and EF. I say that as GC is to AA', so EH is to AA', BC, DD'.

For let FK be made equal to BC, and FL to DD'. And since FK is equal to BC, of which FH is equal to BG, the remainder HK is thus equal to the remainder GC. And since as EF is to DD', so DD' to BC, and BC to AA' [Prop. 7.13], and DD' equal to FL, and BC to FK, and AA' to FH, thus as EF is to FL, so LF to FK, and FK to FH. By separation, as EL to LF, so LK to FK, and KH to FH [Props. 7.11, 7.13]. And thus as one of the leading is to one of the following, so (the sum of) all of the leading to (the sum of) all of the following [Prop. 7.12]. Thus, as KH is to FH, so EL, LK, KH to LF, FK, HF. And KH equal to CG, and FH to AA', and LF, FK, HF to DD', BC, AA'. Thus, as CG is to AA', so EH to DD', BC, AA'. Thus, as the excess of the second is to the first, so is the excess of the last is to all those before it. The very thing it was required to show."

The terseness of Euclid's propositions and proofs may have been a necessity. As is, the Elements of Geometry is over 500 pages of propositions and proofs. Making copies of this popular textbook was labor intensive given that the printing press was not invented until 1440. And the book's popularity lasted a long time: as stated in the cited introduction to an English translation, Elements of Geometry "has the distinction of being the world's oldest continuously used mathematical textbook." So being very terse was being very practical. The proof of Proposition 35 in Book IX could have been even more compact if Euclid could have somehow avoided explicitly equating lengths of specific line segments from different terms in the series. For example, the contemporary notation for geometric series (i.e., a + ar + ar2 + ar3 + ... + arn) does not label specific portions of terms that are equal to each other.

Also in the cited introduction the editor comments, "Most of the theorems appearing in the Elements were not discovered by Euclid himself, but were the work of earlier Greek mathematicians such as Pythagoras (and his school), Hippocrates of Chios, Theaetetus of Athens, and Eudoxus of Cnidos. However, Euclid is generally credited with arranging these theorems in a logical manner, so as to demonstrate (admittedly, not always with the rigour demanded by modern mathematics) that they necessarily follow from five simple axioms. Euclid is also credited with devising a number of particularly ingenious proofs of previously discovered theorems (e.g., Theorem 48 in Book 1)."

To help translate the proposition and proof into a form that uses current notation, a couple modifications are in the diagram. First, the four horizontal line lengths representing the values of the first four terms of a geometric series are now labeled a, ar, ar2, ar3 in the diagram's left margin. Second, new labels A' and D' are now on the first and third lines so that all the diagram's line segment names consistently specify the segment's starting point and ending point.

Here is a phrase by phrase interpretation of the proposition: Similarly, here is a sentence by sentence interpretation of the proof:

Archimedes of Syracuse (c.287 – c.212 BC)

Archimedes used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. His method was to dissect the area into an infinite number of triangles.

Archimedes' Theorem states that the total area under the parabola is 4/3 of the area of the blue triangle.

Archimedes determined that each green triangle has 1/8 the area of the blue triangle, each yellow triangle has 1/8 the area of a green triangle, and so forth.

Assuming that the blue triangle has area 1, the total area is an infinite sum:
 * $$1 \,+\, 2\left(\frac{1}{8}\right) \,+\, 4\left(\frac{1}{8}\right)^2 \,+\, 8\left(\frac{1}{8}\right)^3 \,+\, \cdots.$$

The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives
 * $$1 \,+\, \frac{1}{4} \,+\, \frac{1}{16} \,+\, \frac{1}{64} \,+\, \cdots.$$

This is a geometric series with common ratio 1/4 and the fractional part is equal to
 * $$\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. $$

The sum is
 * $$\frac{1}{1 -r}\;=\;\frac{1}{1 -\frac{1}{4}}\;=\;\frac{4}{3}.$$

This computation uses the method of exhaustion, an early version of integration. Using calculus, the same area could be found by a definite integral.

Nicole Oresme (c.1323 – 1382)

Among his insights into infinite series, in addition to his elegantly simple proof of the divergence of the harmonic series, Nicole Oresme proved that the series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. His diagram for his geometric proof, similar to the adjacent diagram, shows a two dimensional geometric series. The first dimension is horizontal, in the bottom row showing the geometric series S = 1/2 + 1/4 + 1/8 + 1/16 + ..., which is the geometric series with coefficient a = 1/2 and common ratio r = 1/2 that converges to S = a / (1-r) = (1/2) / (1-1/2) = 1. The second dimension is vertical, where the bottom row is a new coefficient aT equal to S and each subsequent row above it is scaled by the same common ratio r = 1/2, making another geometric series T = 1 + 1/2 + 1/4 + 1/8 + ..., which is the geometric series with coefficient aT = S = 1 and common ratio r = 1/2 that converges to T = aT / (1-r) = S / (1-r) = a / (1-r) / (1-r) = (1/2) / (1-1/2) / (1-1/2) = 2.

Although difficult to visualize beyond three dimensions, Oresme's insight generalizes to any dimension d. Using the sum of the d−1 dimension of the geometric series as the coefficient a in the d dimension of the geometric series results in a d-dimensional geometric series converging to Sd / a = 1 / (1-r)d within the range |r|<1. Pascal's triangle and long division reveals the coefficients of these multi-dimensional geometric series, where the closed form is valid only within the range |r|<1.

 * $$\begin{matrix}

1 \\               1 \quad 1 \\ 1 \quad 2 \quad 1 \\ 1 \quad 3 \quad 3 \quad  1 \\ 1 \quad 4 \quad 6 \quad 4 \quad 1 \end{matrix}$$

As an aside, instead of using long division, it is also possible to calculate the coefficients of the d-dimensional geometric series by integrating the coefficients of dimension d−1. This mapping from division by 1-r in the power series sum domain to integration in the power series coefficient domain is a discrete form of the mapping performed by the Laplace transform. MIT Professor Arthur Mattuck shows how to derive the Laplace transform from the power series in this lecture video, where the power series is a mapping between discrete coefficients and a sum and the Laplace transform is a mapping between continuous weights and an integral.

Economics
In economics, geometric series are used to represent the present value of an annuity (a sum of money to be paid in regular intervals).

For example, suppose that a payment of $100 will be made to the owner of the annuity once per year (at the end of the year) in perpetuity. Receiving $100 a year from now is worth less than an immediate $100, because one cannot invest the money until one receives it. In particular, the present value of $100 one year in the future is $100 / (1 + $$I$$ ), where $$I$$ is the yearly interest rate.

Similarly, a payment of $100 two years in the future has a present value of $100 / (1 + $$I$$)2 (squared because two years' worth of interest is lost by not receiving the money right now). Therefore, the present value of receiving $100 per year in perpetuity is

 * $$\sum_{n=1}^\infty \frac{\$100}{(1+I)^n},$$

which is the infinite series:

 * $$\frac{\$ 100}{(1+I)} \,+\, \frac{\$ 100}{(1+I)^2} \,+\, \frac{\$ 100}{(1+I)^3} \,+\, \frac{\$ 100}{(1+I)^4} \,+\, \cdots.$$

This is a geometric series with common ratio 1 / (1 + $$I$$ ). The sum is the first term divided by (one minus the common ratio):

 * $$\frac{\$ 100/(1+I)}{1 - 1/(1+I)} \;=\; \frac{\$ 100}{I}.$$

For example, if the yearly interest rate is 10% ($$I$$ = 0.10), then the entire annuity has a present value of $100 / 0.10 = $1000.

This sort of calculation is used to compute the APR of a loan (such as a mortgage loan). It can also be used to estimate the present value of expected stock dividends, or the terminal value of a financial asset assuming a stable growth rate.

Fractal geometry
The area inside the Koch snowflake can be described as the union of infinitely many equilateral triangles (see figure). Each side of the green triangle is exactly 1/3 the size of a side of the large blue triangle, and therefore has exactly 1/9 the area. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking the blue triangle as a unit of area, the total area of the snowflake is

 * $$1 \,+\, 3\left(\frac{1}{9}\right) \,+\, 12\left(\frac{1}{9}\right)^2 \,+\, 48\left(\frac{1}{9}\right)^3 \,+\, \cdots.$$

The first term of this series represents the area of the blue triangle, the second term the total area of the three green triangles, the third term the total area of the twelve yellow triangles, and so forth. Excluding the initial 1, this series is geometric with constant ratio r = 4/9. The first term of the geometric series is a = 3(1/9) = 1/3, so the sum is

 * $$1\,+\,\frac{a}{1-r}\;=\;1\,+\,\frac{\frac{1}{3}}{1-\frac{4}{9}}\;=\;\frac{8}{5}.$$

Thus the Koch snowflake has 8/5 of the area of the base triangle.

Integration
The derivative of $$f(x) = \arctan(u(x)) \text{ is } f'(x) = u'(x)/(1+[u(x)]^2)$$ because, letting $$y \text{ and }u \text{ represent } f(x) \text{ and } u(x),$$

\begin{align} y &= \arctan(u) &&\quad \text{ implies } \\ u &= \tan(y) &&\quad \text{ in the range } -\pi/2 < y < \pi/2 \text{ and } \\ u' &= \sec^2y \cdot y' &&\quad \text{ by applying the quotient rule to } \tan(y) = \sin(y)/\cos(y), \\ y' &= u'/\sec^2y &&\quad \text{ by dividing both sides by } \sec^2y, \\ &= u'/(1+\tan^2y) &&\quad \text{ by using the trigonometric identity derived by dividing } \sin^2y + \cos^2y = 1 \text{ by } \cos^2y, \\ &= u'/(1+u^2) &&\quad \text{ by recalling } u = \tan(y). \end{align} $$

Therefore, letting $$ u(x) = x, \arctan(x) $$ is the integral

\begin{align} \arctan(x)&=\int\frac{dx}{1+x^2} \quad &&\text{in the range } -\pi/2 < \arctan(x) < \pi/2,\\ &=\int\frac{dx}{1-(-x^2)} \quad &&\text{by writing integrand as closed form of geometric series with }r = -x^2,\\ &=\int\left(1 + \left(-x^2\right) + \left(-x^2\right)^2 + \left(-x^2\right)^3+\cdots\right)dx \quad &&\text{by writing geometric series in expanded form},\\ &=\int\left(1-x^2+x^4-x^6+\cdots\right)dx \quad &&\text{by calculating the sign and power of each term in integrand},\\ &=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots \quad &&\text{by integrating each term},\\ &=\sum^{\infty}_{n=0} \frac{(-1)^n}{2n+1} x^{2n+1} \quad &&\text{by writing series in generator form}, \end{align} $$

which is called Gregory's series and is commonly attributed to Madhava of Sangamagrama (c. 1340 – c. 1425).

Instances

 * : 1 − 1 + 1 − 1 + ⋯
 * A geometric series is a unit series (the series sum converges to one) if and only if |r| < 1 and a + r = 1 (equivalent to the more familiar form S = a / (1 - r) = 1 when |r| < 1). Therefore, an alternating series is also a unit series when -1 < r < 0 and a + r = 1 (for example, coefficient a = 1.7 and common ratio r = -0.7).
 * The terms of a geometric series are also the terms of a generalized Fibonacci sequence (Fn = Fn-1 + Fn-2 but without requiring F0 = 0 and F1 = 1) when a geometric series common ratio r satisfies the constraint 1 + r = r2, which according to the quadratic formula is when the common ratio r equals the golden ratio (i.e., common ratio r = (1 ± √5)/2).
 * The only geometric series that is a unit series and also has terms of a generalized Fibonacci sequence has the golden ratio as its coefficient a and the conjugate golden ratio as its common ratio r (i.e., a = (1 + √5)/2 and r = (1 - √5)/2). It is a unit series because a + r = 1 and |r| < 1, it is a generalized Fibonacci sequence because 1 + r = r2, and it is an alternating series because r < 0.
 * A geometric series is a unit series (the series sum converges to one) if and only if |r| < 1 and a + r = 1 (equivalent to the more familiar form S = a / (1 - r) = 1 when |r| < 1). Therefore, an alternating series is also a unit series when -1 < r < 0 and a + r = 1 (for example, coefficient a = 1.7 and common ratio r = -0.7).
 * The terms of a geometric series are also the terms of a generalized Fibonacci sequence (Fn = Fn-1 + Fn-2 but without requiring F0 = 0 and F1 = 1) when a geometric series common ratio r satisfies the constraint 1 + r = r2, which according to the quadratic formula is when the common ratio r equals the golden ratio (i.e., common ratio r = (1 ± √5)/2).
 * The only geometric series that is a unit series and also has terms of a generalized Fibonacci sequence has the golden ratio as its coefficient a and the conjugate golden ratio as its common ratio r (i.e., a = (1 + √5)/2 and r = (1 - √5)/2). It is a unit series because a + r = 1 and |r| < 1, it is a generalized Fibonacci sequence because 1 + r = r2, and it is an alternating series because r < 0.
 * The terms of a geometric series are also the terms of a generalized Fibonacci sequence (Fn = Fn-1 + Fn-2 but without requiring F0 = 0 and F1 = 1) when a geometric series common ratio r satisfies the constraint 1 + r = r2, which according to the quadratic formula is when the common ratio r equals the golden ratio (i.e., common ratio r = (1 ± √5)/2).
 * The only geometric series that is a unit series and also has terms of a generalized Fibonacci sequence has the golden ratio as its coefficient a and the conjugate golden ratio as its common ratio r (i.e., a = (1 + √5)/2 and r = (1 - √5)/2). It is a unit series because a + r = 1 and |r| < 1, it is a generalized Fibonacci sequence because 1 + r = r2, and it is an alternating series because r < 0.

Geometric series

The geometric series has two degrees of freedom: one for its coefficient a and another for its common ratio r. In the map of polynomials, the big red circle represents the set of all geometric series.

Converging geometric series
Only a subset of all geometric series converge. Specifically, a geometric series converges if and only if its common ratio |r| < 1. In the map of polynomials, the red triangle represents the set of converging geometric series and being drawn inside the big red circle representing the set of all geometric series indicates the converging geometric series is a subset of the geometric series.

Repeated decimals
Only a subset of all converging geometric series converge to decimal fractions that have repeated patterns that continue forever (e.g., 0.7777... or 0.9999... or 0.123412341234...). In the map of polynomials, the little yellow triangle represents the set of geometric series that converge to infinitely repeated decimal patterns. It is drawn inside the red triangle to indicate it is a subset of the converging geometric series, which in turn is drawn inside the big red circle indicating both the converging geometric series and the geometric series that converge to infinitely repeated patterns are subsets of the geometric series.

Although fractions with infinitely repeated decimal patterns can only be approximated when encoded as floating point numbers, they can always be defined exactly as the ratio of two integers and those two integers can be calculated using the geometric series. For example, the repeated decimal fraction 0.7777... can be written as the geometric series

 * $$0.7777\ldots \;=\; \frac{7}{10} \,+\, \frac{7}{10}\frac{1}{10} \,+\, \frac{7}{10}\frac{1}{10^2} \,+\, \frac{7}{10}\frac{1}{10^3} \,+\, \cdots$$

where coefficient a = 7/10 and common ratio r = 1/10. The geometric series closed form reveals the two integers that specify the repeated pattern:

 * $$0.7777\ldots \;=\; \frac{a}{1-r} \;=\; \frac{7/10}{1-1/10} \;=\; \frac{7/10}{9/10} \;=\; \frac{7}{9}.$$

This approach extends beyond base-ten numbers. In fact, any fraction that has an infinitely repeated pattern in base-ten numbers also has an infinitely repeated pattern in numbers written in any other base. For example, looking at the floating point encoding for the number 0.7777...

reveals the binary fraction 0.110001110001110001... where the binary pattern 0b110001 repeats indefinitely and can be written in mostly (except for the powers) binary numbers as

 * $$0.110001110001110001\ldots \;=\; \frac{110001}{1000000} \,+\, \frac{110001}{1000000}\frac{1}{1000000} \,+\, \frac{110001}{1000000}\frac{1}{1000000^2} \,+\, \frac{110001}{1000000}\frac{1}{1000000^3} \,+\, \cdots$$

where coefficient a = 0b110001 / 0b1000000 = 49 / 64 and common ratio r = 1 / 0b1000000 = 1 / 64. Using the geometric series closed form as before

 * $$0.7777\ldots \;=\; 0b0.110001110001110001\ldots \;=\; \frac{a}{1-r} \;=\; \frac{49/64}{1-1/64} \;=\; \frac{49/64}{63/64} \;=\; \frac{49}{63} \;=\; \frac{7}{9}.$$

You may have noticed that the floating point encoding does not capture the 0b110001 repeat pattern in the last couple (least significant) bits. This is because floating point encoding rounds the remainder instead of truncating it. Therefore, if the most significant bit of the remainder is 1, the least significant bit of the encoded fraction gets incremented and that will cause a carry if the least significant bit of the fraction is already 1, which can cause another carry if that bit of the fraction is already a 1, which can cause another carry, etc. This floating point rounding and the subsequent carry propagation explains why the floating point encoding for 0.99999... is exactly the same as the floating point encoding for 1.

As an example that has four digits in the repeated pattern, 0.123412341234... can be written as the geometric series

 * $$0.123412341234\ldots \;=\; \frac{1234}{10000} \,+\, \frac{1234}{10000}\frac{1}{10000} \,+\, \frac{1234}{10000}\frac{1}{10000^2} \,+\, \frac{1234}{10000}\frac{1}{10000^3} \,+\, \cdots$$

where coefficient a = 1234/10000 and common ratio r = 1/10000. The geometric series closed form reveals the two integers that specify the repeated pattern:

 * $$0.123412341234\ldots \;=\; \frac{a}{1-r} \;=\; \frac{1234/10000}{1-1/10000} \;=\; \frac{1234/10000}{9999/10000} \;=\; \frac{1234}{9999}.$$

Power series
Like the geometric series, the power series has one degree of freedom for its common ratio r (along the x-axis) but has n+1 degrees of freedom for its coefficients (along the y-axis), where n represents the power of the last term in the partial series. In the map of polynomials, the big blue circle represents the set of all power series.

Binary encoded numbers
Zeno of Elea's geometric series with coefficient a=1/2 and common ratio r=1/2 is the foundation of binary encoded approximations of fractions in digital computers. Concretely, the geometric series written in its normalized vector form is s/a = [1 1 1 1 1 …][1 r r2 r3 r4 …]T. Keeping the column vector of basis functions [1 r r2 r3 r4 …]T the same but generalizing the row vector [1 1 1 1 1 …] so that each entry can be either a 0 or a 1 allows for an approximate encoding of any fraction. For example, the value v = 0.34375 is encoded as v/a = [0 1 0 1 1 0 …][1 r r2 r3 r4 …]T where coefficient a = 1/2 and common ratio r = 1/2. Typically, the row vector is written in the more compact binary form v = 0.010110 which is 0.34375 in decimal.

Similarly, the geometric series with coefficient a=1 and common ratio r=2 is the foundation for binary encoded integers in digital computers. Again, the geometric series written in its normalized vector form is s/a = [1 1 1 1 1 …][1 r r2 r3 r4 …]T. Keeping the column vector of basis functions [1 r r2 r3 r4 …]T the same but generalizing the row vector [1 1 1 1 1 …] so that each entry can be either a 0 or a 1 allows for an encoding of any integer. For example, the value v = 151 is encoded as v/a = [1 1 1 0 1 0 0 1 0 …][1 r r2 r3 r4 r5 r6 r7 r8 …]T where coefficient a = 1 and common ratio r = 2. Typically, the row vector is written in reverse order (so that the most significant bit is first) in the more compact binary form v = …010010111 = 10010111 which is 151 in decimal.

As shown in the adjacent figure, the standard binary encoding of a 32-bit floating point number is a combination of a binary encoded integer and a binary encoded fraction, beginning at the most significant bit with
 * the sign bit, followed by
 * an 8-bit integer exponent field with an assumed offset of 127 (so a value of 127 represents an exponent value of 0) and with a base of 2 meaning that the exponent value specifies a bit shift of the fraction field, followed by
 * a 23-bit fraction field with an assumed but not encoded 1 serving as the fraction's most significant nonzero bit which would be in bit position 23 if it were encoded.

Building upon the previous example of 0.34375 having binary encoding of 0.010110, a floating point encoding (according to the IEEE 754 standard) of 0.34375 is
 * the sign bit which is 0 because the number is not negative,
 * an 8-bit integer exponent field which must specify a shift that counters the 2 bit left shift to get the original binary encoding from 0.010110 to 1.0110, and that counter shift to recover the original binary encoding is a right shift of 2 bits which is specified by an exponent value of 125 (because 125 − 127 = -2 which is a right shift of 2 bits) which in binary is 0111 1101,
 * a 23-bit fraction field: .0110 0000 0000 0000 0000 000.

Although encoding floating point numbers by hand like this is possible, letting a computer do it is easier and less error prone. The following Julia code confirms the hand calculated floating point encoding of the number 0.34375:

Complex Fourier series

As an example of the ability of the complex Fourier series to trace any 2D closed figure, in the adjacent animation a complex Fourier series traces the letter 'e' (for exponential). Given the intricate coordination of motions shown in the animation, a definition of the complex Fourier series can be surprisingly compact in just two equations:

 * $$\begin{align}

s(t) &= \sum_{n=-\infty}^\infty c_n e^{2 \pi int} \\ c_n &= \int_{0}^1 s(t) e^{-2\pi int} dt \\ \end{align}$$

where parameterized function s(t) traces some 2D closed figure in the complex plane as the parameter t progresses through the period from 0 to 1.

To help make sense of these compact equations defining the complex Fourier series, note that the complex Fourier series summation looks similar to the complex geometric series except that the complex Fourier series is basically two complex geometric series (one set of terms rotating in the positive direction and another set of terms rotating in the negative direction), and the coefficients of the complex Fourier series are complex constants that can vary from term to term. By allowing terms to rotate in either direction, the series becomes capable of tracing any 2D closed figure. In contrast, the complex geometric series has all the terms rotating in the same direction and it can trace only circles. Allowing the coefficients of the complex geometric series to vary from term to term would expand upon the shapes it can trace but all the possible shapes would still be limited to being puffy and cloud-like, not able to trace the shape of a simple line segment, for example going back and forth between 1 + i0 and -1 + i0. However, Euler's formula shows that the addition of just two terms rotating in opposite directions can trace that line segment between 1 + i0 and -1 + i0:

 * $$\begin{align}

e^{i\theta} &= \cos\theta + i\sin\theta \\ e^{-i\theta} &= \cos\theta - i\sin\theta \\ \cos\theta &= \frac{e^{i\theta} + e^{-i\theta}}{2}. \\ \end{align}$$

Concerning the complex Fourier series second equation defining how to calculate the coefficients, the coefficient of the non-rotating term c0 can be calculated by integrating the complex Fourier series first equation over the range of one period from 0 to 1. Over that range, all the rotating terms integrate to zero, leaving just c0. Similarly, any of the terms in the complex Fourier series first equation can be made to be a non-rotating term by multiplying both sides of the equation by $$e^{-2\pi int}$$ before integrating to calculate cn, and that is the complex Fourier series second equation.

History and philosophy

 * C. H. Edwards Jr. (1994). The Historical Development of the Calculus, 3rd ed., Springer.  ISBN 978-0-387-94313-8.
 * Eli Maor (1991). To Infinity and Beyond: A Cultural History of the Infinite, Princeton University Press. ISBN 978-0-691-02511-7
 * Morr Lazerowitz (2000). The Structure of Metaphysics (International Library of Philosophy), Routledge. ISBN 978-0-415-22526-7
 * Morr Lazerowitz (2000). The Structure of Metaphysics (International Library of Philosophy), Routledge. ISBN 978-0-415-22526-7

Economics

 * Carl P. Simon and Lawrence Blume (1994). Mathematics for Economists, W. W. Norton & Company.  ISBN 978-0-393-95733-4
 * Mike Rosser (2003). Basic Mathematics for Economists, 2nd ed., Routledge. ISBN 978-0-415-26784-7

Biology

 * Edward Batschelet (1992). Introduction to Mathematics for Life Scientists, 3rd ed., Springer.  ISBN 978-0-387-09648-3
 * Richard F. Burton (1998). Biology by Numbers: An Encouragement to Quantitative Thinking, Cambridge University Press.  ISBN 978-0-521-57698-7

Computer science

 * John Rast Hubbard (2000). Schaum's Outline of Theory and Problems of Data Structures With Java, McGraw-Hill. ISBN 978-0-07-137870-3

title: Geometric series


text:
Talk:Gypsy (musical)

Various edits
I want to clarify some of my thinking, as people have put in a lot of work and thought to this article, and I don't want anyone to think that I haven't been carefully reading--and appreciating-- the work.

 * I've put back a sentence into the opening, to show the importance of this play--it just seems to fit, right up front. The trivia section does not read very well, it is just a collection of bits and pieces, I'd like to see the items integrated. I removed the sentence about amateur productions as being somewhat irrelevant. The item on being in a book is not very useful, Gypsy is in many many books.

 * The sentence in trivia: "The role of Mamma Rose is seen as one of the biggest ever written for the stage." is certainly true, but perhaps can be expanded, or added onto the Frank Rich statement about "King Lear". As it stands, it is perhaps too self-evident, almost a little bit close to point-of-view.
 * I've added some specific comments about the character of "Rose" to replace the more general statement noted above (in Trivia), and removed the sentence from Trivia.JeanColumbia 12:35, 18 March 2007 (UTC)

 * We use, variously, "Mama Rose", "Mamma Rose", and "Rose". I've even seen her referred to as "Madame Rose". I'm beginning to prefer just "Rose". She is listed that way in the playbill of the 2003 revival and on the Internet Broadway Database. Until someone gets a script, I'm in favor of "Rose".
 * Today I looked at the printed script on Google books--it does not seem to refer to her by any other name but "Rose". However, enough critics (Walter Kerr, Frank Rich) do refer to her as "Mama Rose" so that I will not worry about this. JeanColumbia 12:35, 18 March 2007 (UTC)

 * Using the heading "Patti Lupone Producton" really doesn't seem logical to me, unless we use, uniformly, the "Ethel Merman Production", etc. Probably some writers do that, in newspaper and magazine articles, but I've not seen that usage in Wikipedia for other plays that have revivals. (This has nothing to do with Ms. LuPone but rather terminology.) —The preceding unsigned comment was added by JeanColumbia (talk • contribs) 00:31, 8 March 2007 (UTC).

Sorry I'm late to this talk page, but I agree with the conclusions that you came to. -- Ssilvers 03:34, 20 April 2007 (UTC)

Trivia
The Wikpedia article on trivia states: "This page in a nutshell: Lists of facts, as found in trivia sections, are better presented within the context of the text rather than in a section of unrelated items." reference: WP:TRIV. JeanColumbia 12:01, 10 March 2007 (UTC)

 * I agree. If the facts are of interest, they should go either in "background/history", "productions" or a "Cultural impact" section.  -- Ssilvers 01:55, 20 April 2007 (UTC)

Split musical and film articles
Both the musical and film are notable, I would think, and the film article ought to be split off of the musical article. Then the film project tag should be moved to the film article's talk page (also the "what links here" links pertaining to the film will need to be moved. -- Ssilvers 01:55, 20 April 2007 (UTC)

 * OK, sounds good. Seems simple enough: 1)take the film and tv section out of this article and place it in the existing Gypsy film article 2) may need to make changes to the film article, for consistency, redundancies, etc 3) remove film stub from this article and place it in the film article 4) do the links.

 * I'll have some time either Friday evening or Sunday and could try to do this, if no one else has the time. (Also probably a good idea to wait to see if anyone would like to discuss further.)JeanColumbia 02:57, 20 April 2007 (UTC)

 * Oops! Sorry, I didn't realize there was already a film article!  I was fooled by the film project tag on this page.  Once there is a separate film article, the project tag is supposed to come off.  Sorry, I don't think there's anything to move over.  I just slimmed down the info on the move a little bit.  I don't mind keeping what's there now, as it reflects on the notability of the musical and its influence on other media.  Feel free to do step 2) of your list above and check on 4, but supposedly 4 was already done when the film article moved over.  Best regards, -- Ssilvers 03:34, 20 April 2007 (UTC)

Now that I've embarrassed myself by not paying attention to the disambiguation notice, let me raise another issue. The article needs information about the genesis/background of the show's creation. But that'll have to wait for another day. Best regards, -- Ssilvers 03:57, 20 April 2007 (UTC)

→OK, film/tv/split resolved--I'll look again at the film article, but I think it's probably OKJeanColumbia 11:11, 20 April 2007 (UTC)

Genesis/Background
I've often thought it would be good to have the quite interesting background of this play/musical in the article--but I just don't know how to write it. I have some of the reference books I think I need to refer to, and have read one or two others. I think I could do a first draft, perhaps Sunday or Monday, if no one else wants to. I need to re-read some material, perhaps get the Arthur Laurents book again.JeanColumbia 11:11, 20 April 2007 (UTC)

Notes on "Background"

 * I took a deep breath and wrote the "Background". I decided to keep it short, just showing, in effect, who wanted who, trying to show the influences between the principals. There's a wealth of material about rehearsals, out-of-town tryouts, etc, and if anyone wants that material, it could certainly be added. My main source is, as I referenced, the Zadan "Sondheim & Co."

→Feedback, discussion?JeanColumbia 14:13, 20 April 2007 (UTC)

 * I think it's very nicely done. There may be more to say that can be gleaned from other sources, but this is a good, brief explanation and adds a lot to the article.  -- Best regards, -- Ssilvers 15:13, 20 April 2007 (UTC)

Article structure
As I write this (10:37am (EDT), April 22, the film and tv section has been moved to the very end of the article, after references, and external links. I suspect this was not meant to happen. Therefore, I am moving that section to immediately after "Stage Productions" (which is where the section has been).

Should anyone want this section to be at the very end of the article (after "references" and "external links"), your choice.JeanColumbia 14:42, 22 April 2007 (UTC)
 * Turned out to be a minor edit situation, easily resolved (a code " " had been dropped somewhere along the line). Case closed.JeanColumbia 14:48, 22 April 2007 (UTC)

Awards
I wonder if we're going overboard with the awards. Is it necessary to list the Theatre World awards? Best regards, -- Ssilvers (talk) 15:37, 18 December 2007 (UTC)
 * I think so (since I did it!). Those awards are important because they are sometimes the first award for the new performer, and I think a "first" is important. If you folks think they are not important, best to include that specifically in the Musical Theatre structure article (and then get one of the programming whizes to make a program to delete them from all of the articles they are in.) I shall not put any others in, but I will not remove them, awaiting the consensus. (Just occurred to me: 1) would you also question the listing of the Theatre World Award on an individual performers' article? 2) Would you delete the entire article on "Theatre World Award" as being not notable?) JeanColumbia (talk) 16:15, 18 December 2007 (UTC)

Hello, Jean. As far as I can tell, only you, me and MusicMaker are really active in the project (although other editors have an interest in particular musicals). If you think these awards are important to a general reader of the encyclopedia, then I will rely on your excellent judgement. It just seemed to me that these award sections are rather long in relation to the length of most articles on musicals. Perhaps columnarization would help? BTW, if you think any changes ought to be made to the sturcture guidelines, please propose them on the Musicals project talk page. Since you are one of the most active editors in the project at present, your experience as to what ought to be included or excluded from our articles is very important, and I think that the structure guidelines would benefit from your advice. Best regards, and happy editing! -- Ssilvers (talk) 17:30, 18 December 2007 (UTC)
 * Hi, Ssilvers--OK, let's leave Theatre World Awards in, at least for now, but I must admit I don't care for the look (and readability) of some of the long Awards/nominations sections. I don't have time right now, but shortly I'll try out some other formats--columns are good, probably the best I can think of off-hand. JeanColumbia (talk) 17:37, 18 December 2007 (UTC)

Reformatting
Took a stab at it. Thoughts? -- omtay 38  20:47, 19 December 2007 (UTC)

 * Well, it looks great - you're an html wiz! But I think it is harder to see, at a glance, who won and who was nominated.  I can't tell, for the original productions, if all of those were wins or nominations.  How about putting winners first in bold and then nominees in *not* bold, and then the footnote could clarify that winners are in bold and other nominees in not bold?  -- Ssilvers (talk) 21:40, 19 December 2007 (UTC)

 * This actually motivated me to create a more general format for all these large award sections popping up in articles. I've made a mockup here. Take a look and tell me what you think! (And yah, I know it's a little hard to read, that's been my biggest concern so far.) I used a massive example over there to see if it would work and it also is hard to read. But take a look, tell me what you think! -- omtay 38  22:05, 19 December 2007 (UTC)

 * Mwah!! to Mr. Omtay38! Keep trying, looks very interesting so far. 1) I agree, it's a little hard to tell winners from nominees (personally I don't care much for footnotes, I like it to pop out); 2) can an ordinary mortal make or add to or alter one of these tables (for example, a semi-computer literate Medicare-eligible senior? No one get insulted, that's ME...)JeanColumbia (talk) 22:34, 19 December 2007 (UTC)

 * How about a pre-note? (if i put the note at the top of the table instead of the bottom). As for the ease of formatting, I'm hoping to figure that one out after we get a nice, visually pleasing, format down. If all goes as planned, it should be as easy to ad as any normal infobox. It should.... :-) -- omtay 38  22:38, 19 December 2007 (UTC)

2008 production cast; WP:CRYSTAL
I think it's premature to add the 2008 cast to the table, because one or more of them could get sick prior to the opening, the revival itself could get cancelled, and a whole host of things could go wrong. Look at what just happened with Lone Star Love. Just my opinion. -- Ssilvers (talk) 17:08, 10 January 2008 (UTC)
 * I'm laughing here--Lone Star Love! I don't think it's premature, they are selling tickets as of yesterday. I can imagine many things that could go wrong in life, but this not going on with Patti/Arthur is not one of them (a bit of a joke). However, I don't care, will defer to anyone on this. (At least I got practice in adding to the chart--not something I'm good at.) JeanColumbia (talk) 17:18, 10 January 2008 (UTC)
 * As requested, I removed the 2008 Broadway revival cast from the cast list chart. (Note that I am removing my own work.) JeanColumbia (talk) 13:48, 11 January 2008 (UTC)

Song list
I reverted (twice) the edits to this song list, on 4/23 & 24/08. The newly added edits seemed to be in non-standard format (using ... instead of "—"), added odd characters, and showing "MONTAGE" (that may be on stage but is not part of the song list); adding reprises, etc. I prefer to stay with a reliable source WP:RS (IBDB) (which is WP:VERIFY). This is a well-established song list, I cannot see any reason to think it would change! JeanColumbia (talk) 23:25, 24 April 2008 (UTC)

 * I agree. The changes seem to be erroneous.  -- Ssilvers (talk) 05:04, 25 April 2008 (UTC)

I've looked at the IBDB listing for the songs for the 2008 revival and compared with the last few revivals; of course I am wrong, this "well-established song list" ((my words) did indeed change. I've added the changes as notes to the song list. JeanColumbia (talk) 11:32, 25 April 2008 (UTC)

Analysis; critical response
There was too much analysis of Rose in the Lead, so I moved it down. Eventually, there should be a Critical response section and an analysis section that describes the reviews and gives more analysis of the show and its characters. Best regards, -- Ssilvers (talk) 16:08, 6 May 2008 (UTC)
 * Good, I wanted to move it but I couldn't figure out where to put it. (The Frank Rich article I referenced (from May 2003) gives a nice analysis of the show, for future reference. I also find the Martin Gottfried description and analysis in his book "Sondheim" useful.) By the way, when the Tonys are announced next week, will someone be willing to add the "2008 Revival" award nominations to the Award table?JeanColumbia (talk) 16:19, 6 May 2008 (UTC)

Urk! Not me! I'm hopeless with coding. If you set up the table, I suppose I could try to help insert the info without messing up your pretty table! -- Ssilvers (talk) 16:33, 6 May 2008 (UTC)
 * I think we've got it looking good (and, thanks to omtay38 for the pretty table). I added a few names to the Drama Desks. I don't know how this would look with "small", but I'll fool around with it later, no more time right now. JeanColumbia (talk) 19:39, 6 May 2008 (UTC)

Productions
To answer the question (is this the 1st production in London--1973) more fully. I use as basic references these books: Craig Zadan, "Sondheim & Company", 2nd Edition, 1986; Martin Gottfried, "Sondheim", 1993, 2000; Merle Secrest, "Stephen Sondheim, A Life", 1998, and. Both the web site (sondheimguide.com), the Zadan (pp. 53-54) and Secrest (p. 140) confirm that the 1973 production was the first production in London. I don't see any mention of any other productions in the Gottfried. I think that is sufficient per WP:VERIFY. JeanColumbia (talk) 13:06, 8 May 2008 (UTC)

 * I agree - sorry, I asked too quickly, as my google search confirmed that this was the "long awaited" first production in London. The reviews at Amazon for the cast album are funny, if you want a chuckle.  -- Ssilvers (talk) 13:14, 8 May 2008 (UTC)

Songs recorded
Does this information suggest to you that we should say something about recordings of songs from Gypsy?: http://www.sondheimguide.com/gypsyrec.html#Songs  Sometimes we say something like "Songs of [x musical] have been covered frequently, most notably by [x performer/band] and in [x film/TV show, etc]." -- Ssilvers (talk) 13:24, 8 May 2008 (UTC)

The ending
Here is what Frank Rick wrote--and Sondheim said-- about the ending:
 * "Yet the end of Gypsy is not entirely bleak. "Louise comes out a star and forgives her mother," Mr. Sondheim says. "There is hope for her. Rose does confront who she is in 'Rose's Turn.' There is a catharsis. It's not Rodgers and Hammerstein, but you feel maybe the mother and daughter will come to an understanding and maybe triumph over Rose's craziness and Louise's bitterness." Reference: New York Times article written by Frank Rich, May 4, 2003: . I think this refutes the recent statement (that I deleted) ("...although the show ends with Louise walking away laughing at Rose").JeanColumbia (talk) 22:59, 20 July 2008 (UTC)

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title: Talk:Gypsy (musical)