text:
List of jam bands

The following is a list of notable jam bands, or bands on the jam-band circuit. Jam band performances often feature extended musical improvisation ("jams") over rhythmic grooves and chord patterns, and long sets of music that cross genre boundaries.

0–9

 * 7 Walkers

A

 * Acoustic Syndicate
 * Al and The Transamericans
 * Allman Brothers Band
 * Amfibian
 * Animal Liberation Orchestra
 * Aquarium Rescue Unit
 * Assembly of Dust
 * Ataxia

B

 * Back Door Slam
 * BadBadNotGood
 * The Band
 * Band of Gypsys
 * Banyan
 * Barefoot Truth
 * Béla Fleck and the Flecktones
 * Benevento/Russo Duo
 * Bernie Worrell & the Woo Warriors
 * Big Something
 * Big Tasty
 * Billy Strings
 * Big Head Todd and the Monsters
 * Billy and the Kids
 * The Big Wu
 * BK3
 * Black Midi
 * The Black Crowes
 * Blackberry Smoke
 * Blues Project
 * Blues Traveler
 * Bobby and the Midnites
 * The Bomb Squad
 * BoomBox
 * The Brakes
 * The Brew
 * The Bridge
 * Buckethead
 * Brothers Past

C

 * The Cardinals
 * Carl Broemel
 * Caravan
 * CBDB
 * Centipede
 * Chris Robinson Brotherhood
 * Clutch
 * The Codetalkers
 * Colonel Claypool's Bucket of Bernie Brains
 * Colonel Les Claypool's Fearless Flying Frog Brigade
 * Consider the Source
 * Country Joe and the Fish

D

 * Dark Star Orchestra
 * Dave Matthews Band
 * David Nelson Band
 * The Dead
 * Dead & Company
 * Deep Banana Blackout
 * Derek and the Dominos
 * Derek Trucks Band
 * The Dirty Dozen Brass Band
 * The Disco Biscuits
 * Dispatch
 * Donavon Frankenreiter
 * Donna Jean Godchaux Band
 * Dopapod
 * DJ Logic
 * Donna the Buffalo
 * Dr. Dan Matrazzo and the Looters
 * The Doobie Brothers

E

 * The Egg
 * Ekoostik hookah
 * El Michels Affair
 * Electric Apricot
 * ELO
 * The Electric Co.
 * EOTO
 * The Expendables

F

 * Family Groove Company
 * Fat Freddy's Drop
 * Freddy Jones Band
 * Fungus Amungus
 * Furthur
 * Future Rock

G

 * G. Love & Special Sauce
 * G-Nome Project
 * Gabe Dixon Band
 * Galactic
 * Garage A Trois
 * Garaj Mahal
 * Ghosts of Jupiter
 * Ghost Light
 * Giant Panda Guerilla Dub Squad
 * Glass Harp
 * God Street Wine
 * Gong
 * Goose
 * Gov't Mule
 * Grateful Dead
 * Grace Potter and the Nocturnals
 * The Grapes
 * Greensky Bluegrass
 * Guerilla Toss

H

 * Heart of Gold Band
 * The Heavy Pets
 * Hootie & the Blowfish
 * Hot Buttered Rum
 * Hot Tuna
 * Hypnotic Clambake

I

 * Infamous Stringdusters
 * Ivan Neville's Dumpstaphunk

J

 * Jack Johnson
 * Jacob Fred Jazz Odyssey
 * Jam Camp
 * Jazz Is Dead
 * Jazz Mandolin Project
 * Jefferson Airplane
 * Jerry Garcia Acoustic Band
 * Jerry Garcia Band
 * JGB
 * The Jimi Hendrix Experience
 * The Jimmy Swift Band
 * Joe Russo's Almost Dead
 * John Brown's Body
 * The John Butler Trio
 * Jupiter Coyote

K

 * King Gizzard & the Lizard Wizard
 * Karl Denson's Tiny Universe
 * Keller Williams
 * Khruangbin
 * Kingfish
 * Kikagaku Moyo
 * Kudzu Kings
 * KVHW

L

 * Lake Trout
 * Leaf Hound
 * Leftover Salmon
 * Legion of Mary
 * Les Claypool
 * Lettuce
 * Little Barrie
 * Little Feat
 * Liquid Soul
 * Lotus

M

 * Maktub
 * Man
 * The Marshall Tucker Band
 * Matisyahu
 * Matt Schofield
 * Max Creek
 * The McLovins
 * Medeski Martin & Wood
 * Medeski Scofield Martin & Wood
 * Mercury Rev
 * Michael Franti & Spearhead
 * Mickey Hart
 * Mike Gordon
 * moe.
 * Mofro
 * Molly Hatchet
 * Moshav
 * The Motet
 * The Mother Hips
 * Moon Taxi
 * Mr. Blotto
 * My Morning Jacket

N

 * New Deal
 * New Grass Revival
 * New Riders of the Purple Sage
 * New Monsoon
 * Nickel Creek
 * North Mississippi All-Stars
 * The New Mastersounds

O

 * Old & In the Way
 * Oteil Burbridge
 * The Other Ones
 * Oysterhead
 * Ozric Tentacles
 * O.A.R.

P

 * Particle
 * Pat McGee Band
 * Paul Butterfield Blues Band
 * Perpetual Groove
 * Phil Lesh and Friends
 * Phish
 * Pigeons Playing Ping Pong
 * Polyphonic Spree
 * Pnuma Trio
 * Pseudopod

Q

 * The Quark Alliance
 * Quicksilver Messenger Service

R

 * The Radiators
 * Railroad Earth
 * RAQ
 * Ratdog
 * Reconstruction
 * Return to Forever
 * Rhythm Devils
 * Robert Randolph and the Family Band
 * Rodrigo Y Gabriela
 * Roster McCabe
 * Rubber Souldiers
 * Rusted Root
 * RX Bandits

S

 * Santana
 * The Samples
 * Schleigho
 * SerialPod
 * Sister Hazel
 * Slightly Stoopid
 * The Slip
 * Soulhat
 * Soulfarm
 * Soulive
 * Soul Rebels Brass Band
 * Sound Tribe Sector 9
 * Spafford
 * Spyro Gyra
 * The Spin Doctors
 * State Radio
 * Steve Kimock Band
 * Stockholm Syndrome
 * Strangefolk
 * The String Cheese Incident
 * Sublime

T

 * Tauk
 * Tea Leaf Green
 * Tedeschi Trucks Band
 * Ten Ton Chicken
 * Ten Years After
 * The Bright Light Social Hour
 * The New Mastersounds
 * Traffic
 * Trey Anastasio Band
 * Trigon
 * Twiddle
 * Traveling Wilburys

U

 * Umoja Orchestra
 * Umphrey's McGee

V

 * Vampire Weekend
 * Vetiver
 * Vida Blue
 * Vulfpeck

W

 * War
 * The Waybacks
 * Ween
 * The Werks
 * Widespread Panic
 * Cory Wong
 * The Word

X

 * Xavier Rudd

Y

 * Yonder Mountain String Band

Z

 * Zac Brown Band
 * Zach Gill
 * Zero
 * Zilla
 * ZOX

title: List of jam bands


text:
Songwriters Hall of Fame

The Songwriters Hall of Fame (SHOF) is an American institution founded in 1969 by songwriter Johnny Mercer, music publisher/songwriter Abe Olman, and publisher/executive Howie Richmond to honor those whose work, represent, and maintain, the heritage and legacy of a spectrum of the most beloved English language songs from the world's popular music songbook. It not only celebrates these established songwriters, but is also involved in the development of new English language songwriting talent through workshops, showcases, and scholarships. There are many programs designed to teach and discover new English language songwriters. Nile Rodgers serves as the organization's chairman.

The Hall of Fame was formed in 1969, and in 2010, an exhibit was put on display online inside the Grammy Museum at L.A. Live in Los Angeles. The Hall has no permanent place of residence, and because the awards are not televised, there would be no other digital recording of the event for posterity.

There are numerous examples of collaborating songwriters being inducted in unison, with each person being considered a separate entrant. The inaugural year featured 120 inductees, many of whom had a professional partnership, such as Rodgers and Hammerstein. Burt Bacharach and Hal David followed in 1972. Betty Comden and Adolph Green were selected in 1980, and Leiber and Stoller were inducted in 1985. John Lennon and Paul McCartney were inducted in 1987. In the same year, Gerry Goffin, Carole King, Barry Mann, and Cynthia Weil were inducted into the SHOF. Motown's Holland-Dozier-Holland team were honored the following year. Elton John and Bernie Taupin were among those chosen in 1992, and the pop music group the Bee Gees had all three brothers inducted in 1994. In 1995, Bob Gaudio and Bob Crewe as well as Gamble and Huff were inducted. John Denver was inducted in 1996. The Eagles' Glenn Frey and Don Henley were co-inductees in 2000. Queen was the first rock band to have all their band members inducted in 2003. Five members of Earth, Wind & Fire were in the class of 2010. Four members of Kool and the Gang were honored in 2018. Through 2019, 461 individuals were inducted into the SHOF.

Due to the COVID-19 pandemic, the 2020 Songwriters Hall of Fame Induction Ceremony was postponed until 2022. The Songwriters Hall of Fame president and CEO, Linda Moran, chose to move the event so that a proper celebration could take place. New 2020 inductees would include Mariah Carey, Chad Hugo, the Isley Brothers, Annie Lennox, Steve Miller, Rick Nowels, William “Mickey” Stevenson, Dave Stewart and Pharrell Williams. Additionally, Jody Gerson of Universal Music Group will be given the Abe Olman Publisher Award and Paul Williams is set to receive the Johnny Mercer Award. On March 8, 2022, the ceremony was officially announced to take place on June 16, 2022, at its longtime location, the Marriott Marquis New York’s Times Square.

Abe Olman Publisher Award
The Abe Olman Publisher Award is given to publishers who have had a substantial number of songs that have become world-renowned and who have helped to further the careers and success of many songwriters.
 * 1983 – Howard S. Richmond
 * 1986 – Leonard Feist
 * 1987 – Lou Levy
 * 1988 – Buddy Killen
 * 1990 – Charles Koppelman & Martin Bandier
 * 1991 – Frank Military & Jay Morgenstern
 * 1992 – Bonnie Bourne
 * 1993 – Berry Gordy
 * 1994 – Buddy Morris
 * 1995 – Al Gallico
 * 1996 – Freddy Bienstock
 * 1997 – Gene Goodman
 * 1998 – Irwin Z. Robinson
 * 1999 – Bill Lowery
 * 2000 – Julian Aberbach
 * 2001 – Ralph Peer
 * 2002 – Edward P. Murphy
 * 2003 – Nicholas Firth
 * 2004 – Les Bider
 * 2005 – Beebe Bourne
 * 2006 – Allen Klein
 * 2007 – Don Kirshner
 * 2008 – Milt Okun
 * 2009 – Maxyne Lang
 * 2010 – Keith Mardak
 * 2012 – Lance Freed
 * 2017 – Caroline Bienstock
 * 2022 – Jody Gerson

Board of Directors Award
The Board of Directors Award is presented to an individual selected by the SHOF Board in recognition of his or her service to the songwriting community and the advancement of popular music.
 * 1986 – Jule Styne
 * 1988 – Stanley Adams
 * 1992 – Edward P. Murphy
 * 1996 – Anna Sosenko & Oscar Brand
 * 1997 – Thomas A. Dorsey

Contemporary Icon Award
The Contemporary Icon Award was established in 2015 to recognize songwriter-artists who attained an iconic status in pop culture.
 * 2015 – Lady Gaga
 * 2019 – Justin Timberlake

Global Ambassador Award
In 2017, Pitbull was presented the Global Ambassador Award. The award is given to an individual "whose music has true worldwide appeal, crossing genre, cultural and national boundaries".

Hal David Starlight Award
First presented in 2004 as the Starlight Award, the prize was renamed in 2006 as the Hal David Starlight Award in honor of the SHOF Chairman's longtime support of young songwriters. Award recipients are gifted songwriters who are at an apex in their careers and are making a significant impact in the music industry via their original songs.
 * 2004 – Rob Thomas (Matchbox Twenty)
 * 2005 – Alicia Keys
 * 2006 – John Mayer
 * 2007 – John Legend
 * 2008 – John Rzeznik (Goo Goo Dolls)
 * 2009 – Jason Mraz
 * 2010 – Taylor Swift
 * 2011 – Drake
 * 2012 – Ne-Yo
 * 2013 – Benny Blanco
 * 2014 – Dan Reynolds (Imagine Dragons)
 * 2015 – Nate Ruess (Fun)
 * 2016 – Nick Jonas
 * 2017 – Ed Sheeran
 * 2018 – Sara Bareilles
 * 2019 – Halsey
 * 2022 - Lil Nas X
 * 2023 - Post Malone
 * 2024 - SZA

Howie Richmond Hitmaker Award
The Howie Richmond Hitmaker Award is given to musical artists who have had a substantial number of hit songs across a lengthy career, and who, according to the Hall of Fame, "recognize the importance of songs and their writers".
 * 1981 – Chuck Berry
 * 1983 – Rosemary Clooney & Margaret Whiting
 * 1990 – Whitney Houston
 * 1991 – Barry Manilow
 * 1995 – Michael Bolton
 * 1996 – Gloria Estefan
 * 1998 – Diana Ross
 * 1999 – Natalie Cole
 * 2000 – Johnny Mathis
 * 2001 – Dionne Warwick
 * 2002 – Garth Brooks
 * 2003 – Clive Davis
 * 2008 – Anne Murray
 * 2009 – Tom Jones
 * 2010 – Phil Ramone
 * 2011 – Chaka Khan
 * 2014 – Doug Morris
 * 2016 – Seymour Stein
 * 2018 – Lucian Grainge

Johnny Mercer Award
The Johnny Mercer Award is the highest honor bestowed by the event. It goes to writers already inducted into the Songwriters Hall of Fame for having established a history of outstanding creative works.
 * 1980 – Frank Sinatra
 * 1981 – Yip Harburg
 * 1982 – Harold Arlen
 * 1983 – Sammy Cahn
 * 1985 – Alan Jay Lerner
 * 1986 – Mitchell Parish
 * 1987 – Jerry Herman
 * 1990 – Jerry Bock & Sheldon Harnick
 * 1991 – Betty Comden & Adolph Green
 * 1992 – Burton Lane
 * 1993 – Jule Styne
 * 1994 – Irving Caesar
 * 1995 – Cy Coleman
 * 1996 – Burt Bacharach & Hal David
 * 1997 – Alan and Marilyn Bergman
 * 1998 – Paul Simon
 * 1999 – Stephen Sondheim
 * 2000 – Jerry Leiber and Mike Stoller
 * 2001 – Billy Joel
 * 2002 – Carole King
 * 2003 – Jimmy Webb
 * 2004 – Stevie Wonder
 * 2005 – Smokey Robinson
 * 2006 – Kris Kristofferson
 * 2007 – Dolly Parton
 * 2008 – Paul Anka
 * 2009 – Holland–Dozier–Holland
 * 2010 – Phil Collins
 * 2011 – Barry Mann & Cynthia Weil
 * 2013 – Elton John & Bernie Taupin
 * 2014 – Kenneth Gamble & Leon Huff
 * 2015 – Van Morrison
 * 2016 – Lionel Richie
 * 2017 – Alan Menken
 * 2018 – Neil Diamond
 * 2019 – Carole Bayer Sager
 * 2022 – Paul Williams
 * 2023 – Tim Rice
 * 2024 - Diane Warren

Patron of the Arts
The Patron of the Arts is presented to influential industry executives who are not primarily in the music business but are great supporters of the performing arts.
 * 1988 – Martin Segal
 * 1989 – Roger Enrico
 * 1990 – Edgar Bronfman Jr.
 * 1991 – Edwin M. Cooperman
 * 1992 – Jonathan Tisch
 * 1993 – Michel Roux
 * 1994 – Philip Dusenberry
 * 1995 – Theodore J. Forstmann
 * 1996 – Sumner Redstone
 * 1997 – Dr. Samuel LeFrak
 * 1998 – David Checketts
 * 1999 – Robert Mondavi
 * 2001 – Iris Cantor
 * 2002 – Stephen Swid
 * 2003 – Martin Bandier
 * 2004 – Michael Goldstein
 * 2005 – Henry Juszkiewicz
 * 2023 - Anthony Mason

Pioneer Award
The Pioneer Award was established in 2012 to recognize the career of a historic creator of an extensive body of musical work that has been a major influence on generations of songwriters.
 * 2012 – Woody Guthrie
 * 2013 – Berry Gordy

Sammy Cahn Lifetime Achievement Award
Named for the former President of the Songwriters Hall of Fame, the Sammy Cahn Lifetime Achievement Award is given to individuals or teams who are recognized as having done a great deal to further the successes of songwriters.
 * 1980 – Ethel Merman
 * 1981 – Tony Bennett
 * 1982 – Dinah Shore
 * 1983 – Willie Nelson
 * 1984 – Benny Goodman
 * 1985 – John Hammond
 * 1987 – Jerry Wexler
 * 1988 – Dick Clark
 * 1989 – Quincy Jones
 * 1990 – B.B. King
 * 1991 – Gene Autry
 * 1992 – Nat King Cole
 * 1993 – Ray Charles
 * 1994 – Lena Horne
 * 1995 – Steve Lawrence & Eydie Gormé
 * 1996 – Frankie Laine
 * 1997 – Vic Damone
 * 1998 – Berry Gordy
 * 1999 – Kenny Rogers
 * 2000 – Neil Diamond
 * 2001 – Gloria & Emilio Estefan
 * 2002 – Stevie Wonder
 * 2003 – Patti LaBelle
 * 2004 – Neil Sedaka
 * 2005 – Les Paul
 * 2006 – Peter, Paul & Mary
 * 2012 – Bette Midler

Abe Olman Scholarship
Abe Olman was an American songwriter and music publisher. He was later director of ASCAP, and a founder of the Songwriters Hall of Fame which, in 1983, named the Abe Olman Publisher Award. In his honor, the Abe Olman Scholarship is given out each year by his family in the interest of encouraging and supporting the careers of young songwriters. The scholarship has been awarded since 1989 to individuals such as Matt Katz-Bohen, John Legend, and Bebe Rexha.

Holly Prize
Created in 2010, the Holly Prize is a tribute to the legacy of Buddy Holly, a SHOF inductee. The award recognizes and supports a new "all-in songwriter" — an exceptionally talented and inspired young musician/singer/songwriter whose work exhibits the qualities of Holly's music: true, great and original. The Holly Prize is administered and juried by the SongHall.
 * 2010 – Laura Warshauer
 * 2011 – Kendra Morris
 * 2012 – Emily King
 * 2013 – Ben Howard
 * 2014 – Jack Skuller
 * 2015 – Jenny O.
 * 2016 – Shun Ng
 * 2017 – Katie Pruitt
 * 2018 – Sylvan Esso
 * 2019 – Adia Victoria

NYU Steinhardt Songwriting Scholar Award
Announced in 2011, along with a collaboration between the Songwriters Hall of Fame and NYU Steinhardt's Department of Music and Performing Arts Professions, the NYU Steinhardt Songwriting Scholar Award is presented to a music composition student whose work holds great potential for success in the field, and embodies the art, craft, individuality and qualities of communication of the best songwriting.

Towering Performance Award
The Towering Performance Award is given in recognition of one-of-a-kind performances by one-of-a-kind singers that have recorded outstanding and unforgettable interpretations of songs that have become iconic standards.
 * 2003 - Tony Bennett
 * 2005 – Bill Medley
 * 2009 – Andy Williams
 * 2012 – Ben E. King

Towering Song Award
The Towering Song Award is given to creators of an individual song that has influenced the culture in a unique way over the years.
 * 1995 – "As Time Goes By" (written by Herman Hupfeld)
 * 1996 – "Happy Birthday To You" (written by Patty Hill and Mildred J. Hill)
 * 1997 – "How High The Moon" (lyrics by Nancy Hamilton and music by Morgan Lewis)
 * 1998 – "The Christmas Song" (written by Robert Wells and Mel Tormé)
 * 1999 – "Fly Me To The Moon" (written by Bart Howard)
 * 2000 – "All of Me" (written by Gerald Marks and Seymour Simons)
 * 2000 – "You Are My Sunshine" (recorded by Jimmie Davis)
 * 2001 – "Let Me Call You Sweetheart" (lyrics by Beth Slater Whitson and music by Leo Friedman)
 * 2002 – "You're a Grand Old Flag" (written by George M. Cohan)
 * 2003 – "I Left My Heart in San Francisco" (written by George Cory and Douglass Cross)
 * 2004 – "What the World Needs Now is Love" (lyrics by Hal David and music by Burt Bacharach)
 * 2005 – "You've Lost That Lovin' Feelin'" (written by Phil Spector, Barry Mann and Cynthia Weil)
 * 2006 – "When the Saints Go Marching In" (lyrics by Katharine Purvis and music by James Milton Black)
 * 2007 - "Unchained Melody" (lyrics by Hy Zaret and music by Alex North)
 * 2008 – "Take Me Out to the Ball Game" (lyrics by Jack Norworth and music by Albert Von Tilzer)
 * 2009 – "Moon River" (lyrics by Johnny Mercer and music by Henry Mancini)
 * 2010 – "Bridge Over Troubled Water" (written by Paul Simon)
 * 2011 – "It Was a Very Good Year" (written by Ervin Drake)
 * 2012 – "Stand by Me" (written by Ben E. King, Jerry Leiber and Mike Stoller)
 * 2013 – "A Change Is Gonna Come" (written by Sam Cooke)
 * 2014 – "Over The Rainbow" (lyrics by E.Y. Harburg and music by Harold Arlen)
 * 2015 – "What a Wonderful World" (written by Bob Thiele and George David Weiss)

Visionary Leadership Award
Created in 2011, The Visionary Leadership Award recognizes members of the Hall of Fame Board of Directors who have made a significant contribution in furthering the ongoing mission of the organization.
 * 2011 – Hal David
 * 2014 – Del Bryant
 * 2015 – John A. Lofrumento
 * 2019 – Martin Bandier

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

title: Songwriters Hall of Fame


text:
Imhoff tank

The Imhoff tank, named for German engineer Karl Imhoff (1876–1965), is a chamber suitable for the reception and processing of sewage. It may be used for the clarification of sewage by simple settling and sedimentation, along with anaerobic digestion of the settled sludge. It consists of an upper chamber in which sedimentation takes place, from which settled solids slide down on the inclined bottom slopes towards a lower chamber in which the sludge accumulates. The two chambers are otherwise unconnected, with the more liquid sewage flowing only through the upper sedimentation chamber and only a slow flow of sludge in the lower digestion chamber. The lower chamber requires separate biogas vents and pipes for the removal of digested sludge, typically after 6–9 months of digestion. The Imhoff tank is in effect a two-story septic tank and retains the septic tank's simplicity while eliminating many of its drawbacks, which largely result from the mixing of fresh sewage and septic sludge in the same chamber.

Typically, well-designed and operated Imhoff tanks are expected to remove suspended solids with an efficiency between 50-70%. Effluent coming out from Imhoff tanks can be either discharged in the environment, sent to a centralized wastewater treatment facility, or sent to constructed wetlands for disinfection and nutrient removal.

As a result of anaerobic digestion of settled sludge, methane, carbon dioxide, hydrogen and hydrogen sulphide are typically formed. While in the past this gas mix used to be exploited for energy production given the relatively high methane content, nowadays gas from Imhoff tanks is typically vented out in the environment. This wastes the energy potential recovery of the technology and increases its carbon footprint, given the high content of methane which has a global warming potential about 25 times larger than the one of carbon dioxide.

Imhoff tanks are being superseded in sewage treatment by plain sedimentation tanks using mechanical methods for continuously collecting the sludge, which is moved to separate digestion tanks. This arrangement permits both improved sedimentation results and better temperature control in the digestion process, leading to a more rapid and complete digestion of the sludge.

A test for settleable solids in water, wastewater and stormwater uses an Imhoff cone, with or without stopcock. The volume of solids is measured after a specified time period at the bottom of a one-liter cone using graduated markings.

title: Imhoff tank


text:
Talk:Geometric series

geometric power series application?
In the article's "Coefficient a" section, I describe the relationship between the geometric series and the power series. In the article's "Applications" section, there is a "geometric power series" application. Is there a reference for a series named the "geometric power series"? Gj7 (talk) 20:51, 26 December 2020 (UTC)

 * The comments about differentiating the geometric series have been moved to the Historic insights section on Nicole Oresme. Therefore, all that remains is a description of Gregory's series. In my opinion, there should also be a Historic insights section on Madhava of Sangamagrama and this description of Gregory's series should then be moved there (and the "geometric power series" subsection should then be deleted). Gj7 (talk) 21:49, 7 June 2021 (UTC)

 * I finally changed this heading name from "geometric power series" to "integration" and I attributed the resulting series to Madhava of Sangamagrama. Gj7 (talk) 17:31, 4 June 2022 (UTC)

Original research?
Last year after reading Thousand Brains - a new theory of intelligence by Jeff Hawkins, I started to think about the importance of a useful mental model when learning math. As described in A Thousand Brains, being actionable and simple and hierarchical are characteristics of a useful mental model. Intending to help high school math students, I posted an image of a map of polynomials with the geometric series at the origin, showing the relative locations of the power series, the Laurent series, and matrix polynomials ... all of which are covered in their own Wikipedia articles.

Today, an anonymous editor removed that map of polynomials and claimed it is original research. Can an image that shows a particular organization of existing content be original research? If so, it seems that most images in Wikipedia articles would also be original research (e.g., during the image upload process there is a checkbox to verify that the person uploading the image is also the image creator). Gj7 (talk) 23:03, 25 July 2022 (UTC)
 * I agree with the anonymous editor. This appears to be original research, only dubiously relevant to the topic, overly promotional for the book, and not sourced by reliable publications in the mathematics literature. I don't think it belongs in this article. —David Eppstein (talk) 23:24, 25 July 2022 (UTC)
 * The quotes from the book were not intended as a promotion, but as a reference about the task of finding an actionable, simple, and hierarchical mental model is a key to gaining expertise in anything, including math. Why would the geometric series being at the origin of the image be only dubiously relevant to the topic? Do you know of a different and better mental model of math that illustrates the relationship between different series?  — Preceding unsigned comment added by Gj7 (talk • contribs) 23:49, 25 July 2022 (UTC)

Mistake in section "Repeating decimals"
The statement "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." is fundamentally wrong.

Infinitely repeating patterns occur when dividing by such a number, which contains a prime factor, that is not part of the base of the number system. That means, finite sequences will occur only for a limited number of divisors (in base 10 only for the primes 2 and 5 (and these powers and combinations; in base 2 only for the prime 2 and its powers). Division by any other primes (or their powers and combinations) will result in infinitely repeating patterns.

The example for 0.777... (which is equal to 7/9) used base 2, where it is also an infinitely repeating binary number, but this is no proof. It is only because base 2 does not include the prime number 3, which is the prime number included in the division (7/3^2). If you consider the same in base 3, 7/9 is written as 0.21, or in base 9 it is 0.7. It will be represented in finite length also in base 12 for example, 0.91 would be the representation. 2001:4C4E:10FE:8A00:5803:4BB7:58AC:9406 (talk) 11:20, 28 October 2022 (UTC)
 * There is an infinitely repeating pattern of zeros following any finite-length representation. These numbers in fact have two infinitely repeating representations, one with an infinitely repeating pattern of zeros and another with an infinitely repeating pattern of the base minus one. —David Eppstein (talk) 15:38, 28 October 2022 (UTC)

Correction for topic 'Mistake in section "Repeating decimals"'
The representation of 7/9 in base 12 is 0.94 (and NOT 0.91). Sorry for the typo, I could not change it after publishing my comment. 2001:4C4E:10FE:8A00:5803:4BB7:58AC:9406 (talk) 11:34, 28 October 2022 (UTC)

Opening sentence is confusing.
"In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms." We can't sum an infinite number of terms. This needs to be broken down along the lines as follows: "Each term of a geometric series is the sum of a finite number of terms of a geometric sequence" or similar. TorlachRush (talk) 19:50, 12 April 2023 (UTC)

title: Talk:Geometric series


text:
Talk:Geometric progression

A question
Can common ratio's be imaginary? Maybe we should include that since the article specifically says they can be negative. —Preceding unsigned comment added by 72.229.188.248 (talk) 02:34, 28 January 2011 (UTC)

Paragraph problem
I don't really agree with the paragaph:
 * One cannot see why the proportion called arithmetical is any more arithmetical than that which is called geometrical, nor why the latter is more geometrical than the former. On the contrary, the primitive idea of geometrical proportion is based on arithmetic, for the notion of ratios springs essentially from the consideration of numbers

That seems a rationalisation based on number, but it is not certain that mathematics started as number.

 * It is "not certain" that it did? Indeed, isn't it certain that it did not?  The notion of real number grew out of geometry!

For example, take a square, double the length of its sides, double again, and again. Clearly the side lengths are in geometric progression; so too are the areas. Take a different square, add 2 to the length of the sides (2 what?), add 2 again, and again. This time the side lengths are in arithmetic progression, but the areas are not. It seems natural to call the former geometric, leaving arithmetic to the latter. In the medieval quadrivium, arithmetic was pure number, geometry was number in space, music number in time, and astronomy number in space and time; but I doubt that was the order in pre-history.--Henrygb 13:13, 21 Mar 2004 (UTC)

Both Geometric sequence and Geometric series deal with the other, so i"m putting them together under Geometric progression and redirecting.

progression, series or sequence?
Why isn't this article called "geometric sequence" but "geometric progression"? I don't find it clear what is meant with progression. --Abdull 13:43, 19 Jun 2005 (UTC)

 * a sequence means that the terms are just followed on after the other. a series means that every term is summed to the next. a prograssion just means that the numbers progress, keep going, withut specifyin if the terms are added or not
 * Cako 20:49, 2 November 2006 (UTC)goldencako

Sequences are multiples or powers of a quantity (eg 'r'), shown as a list of terms separated by COMMAS (a,ar,ar2, etc in example). Series show the SUM of those terms (a + ar + ar2 + ...). 'r' or 'x' is fixed for the calculation of a single series, but varies in practice as the top right hand diagram illustrates, so calling it a fixed number is a bit misleading. When I did Maths 2T as part of my Chemical Engineering degree one of the recommended texts was a little book called 'Sequences and Series' (part of the RKP series of maths texts). It all seemed rather dry at the time. Had it been introduced by way an example such as Arrhenius' greenhouse law (see http://en.wikipedia.org/wiki/Svante_Arrhenius#Greenhouse_effect, and upcoming BBC series) it would have been much more intriguing. Geometric series represent exponential growth or decay, as represented by population dynamics (Malthus) or climate change (Arrhenius). I think that the geometric series explanation should be retained to illustrate the relationship between sequences and series, and the difference between arithmetic (LINEAR) progressions/series and geometric (EXPONENTIAL, ie POWER) progressions. Merrison. Merrison (talk) 18:15, 23 November 2015 (UTC)

Interest Rate bad example
The example for return on capital interest is not correct and misleading. Interest rates are calculated using compound interest rates. If one could get $16000 out of $2000 in 6 years, that would be quite awesome. I am not familiar with wikipedia to fix this example, thanks for taking note of it. --Vastinnocentaims 15:20, 16 August 2005 (UTC)

Mistake in ratio?
I think the exponent in the ratio (second image under Formulae) should be 1 / (n - 1) instead of just n - 1. Anyone can verify this? Aggelos Orfanakos 00:09, August 30, 2005 (UTC)

missing formula
it appears that the current version of the page (on my display)is missing the first formula, namely the mathethematical definition of the sequence itself, before going into the definitions of the scale factor and common ratio

Clarification please?
It's been a few years since I studied this subject, so I came on here to help myself solve a little problem I was having, and found that this article was of little help. Can someone please clarify the geometric series part of the article? What is X? I can tell $$X^n$$ is the Nth term of the geometric sequence, but what is X? Maybe put an example in?--149.135.21.11 14:33, 13 March 2006 (UTC)

Notation

 * $$a_n = a\,r^{n-1} \quad \mbox{where n is an integer such that }n \ge 1$$

In the case of the above, would it not be more elegant to say "$$n\in\mathbb{N}$$" in place of "$$\mbox{where n is an integer such that }n \ge 1$$"?

I'd change it myself, but I wouldn't want to mess around with something if there's a specific reason for that form. --MightyPenguin 11:53, 20 March 2006

 * $$\mathbb{N}$$ can contain zero, depending on the circumstances (see natural number). As it's slightly ambiguous, it's probably better to leave it as is. siafu 21:44, 2 May 2006 (UTC)

Credit for Series Simplification
Who is to be given credit for discovering the geometric series simplification? Gauss? 131.120.10.130 19:34, 19 July 2006 (UTC)

Product
Can anyone add a subsection of the product of a geometric sequence?

should i add this:
Hey, I was wondering if it would be appropriate to add this to the article: Why do we call it a Geometric Series? If someone knows the answer, please add this here.

Product
The product of a geometric sequence can be expressed as

 * $$P = \prod_{i=0}^{n-1} ar^i $$

However, it can also be expressed using only the first and last termes and the total number of terms.

 * $$P=(a_1 \cdot a_2)^{n/2}$$

Which leads to the interesting conclusion, that

 * $$\sqrt[n]{P}=a_{\frac{(n+1)}{2}}$$.

 * If so, should i also include some proofs?
 * Cako 20:48, 2 November 2006 (UTC)goldencako

 * A danger for the first claim above is when n is odd, in which case the question arises as to what $$x^{1/2}$$ is for negative or complex $$x$$. E.g. consider the case where $$n = 1$$ and $$a_1 = -1$$.

 * As for the second claim, it is evidently only defined for the case that n is odd. Even then, it doesn't hold if r is allowed to be non-real: suppose that $$n$$ is 3, then multiplying $$a$$ by cis(2π/3) doesn't change P but does change each term $$a_i$$ (including $$a_{\frac{(n+1)}{2}}$$).  The claim might hold if $$n$$ is odd and $$\sqrt[n]{P}$$ is read as ‘one of the $$n$$'th roots of P’ instead of the usual ‘the principal $$n$$'th root of P’.  (See Nth root.)

 * More generally: The principles behind WP::No original research may apply to such derivations: It's good to cite a reliable external source for such derivations, to give more confidence that the corresponding proof has been checked by an expert. (Purely-mathematical deductions are something of a boundary case for the no-original-research principle; e.g. WP::Attribution says that “straightforward mathematical calculations” don't count as original research.)

 * Pjrm 13:11, 1 April 2007 (UTC)

Interesting
I have someone following me around deleting my comments. Anyone want to comment on the following? --JohnLattier 23:06, 3 November 2006 (UTC)

- 	$$\sum_{k=0}^\infty ar^k = \lim_{n\to\infty}{\sum_{k=0}^{n} ar^k}$$ - 		 - 		 - 	Remember, the infinite sum only converges towards the limit. The limit represents the number to which the sum is convergent. The equal sign is improper. --JohnLattier 05:52, 3 November 2006 (UTC) - 		 - 	I suggest: - 		 - 	$$\sum_{k=0}^\infty ar^k \approx \lim_{n\to\infty}{\sum_{k=0}^{n} ar^k}$$ - 	--JohnLattier 05:55, 3 November 2006 (UTC)

 * Ugh. You're wrong, and this article is not going to incorporate your suggestion. End of story. Melchoir 23:58, 3 November 2006 (UTC)

 * I believe it's inappropriate to make such a response to a proposed change in Wikipedia without giving at least some evidence for your position.
 * (In this case, some evidence is that the Infinite series article claims that the $$\sum_{n=0}^\infty$$ notation specifically means the limit of the sequence of partial sums. Granted, citing a Wikipedia article is not particularly strong evidence for what should go in another Wikipedia article, but it's considerably better than nothing.)
 * Even with strong evidence, it is not “end of story”, because opposing evidence can be given. (E.g. in the current case, the dispute is merely about the meaning of a notation, and one notation can sometimes have different meanings.  Here, the meaning in the case of divergent series in particular has varied over time, and may well continue to evolve.  Maybe the exact proposed change should not be adopted, but maybe some aspect of such an objection could reasonably be incorporated into the relevant article.)
 * If one disagrees with a proposed change, then one's own conviction does count as evidence, but is to be set against the opposing evidence that the proposer apparently believes that the proposed change is a good one. It suffices to give the evidence one has on hand (“I am confident that the proposed claim is false”, possibly giving some reason for the person to trust your judgement if you think that they too are confident of their position).  I understand that one may strongly want to prevent certain changes, but I believe that the response above tends to have the undesirable side effect of dissuading future constructive changes.
 * Pjrm 14:50, 1 April 2007 (UTC)

Constant Factor
Why is there a constant factor "a" in all of these formulae? Isn't it more concise to just say $$\sum_{k=0}^\infty r^k = \frac{1}{1-r}$$ instead of :$$\sum_{k=0}^\infty ar^k = \frac{a}{1-r}$$ ? Dougthebug 23:36, 5 February 2007 (UTC)

Ans: Usually when deriving a formula you use a general form of the equation. You just can't assume things like a = 1. You need to derive equations using      generic values. Sachhidh

I absolutely agree that the a is redundant, and should be removed - it adds nothing to the mathematics and anyone who is well-studied enough to be doing finite sums should be able to extract an a from both sides. I feel like it makes the mathematics clumsy and should not be in an encyclopedia. —Preceding unsigned comment added by 165.124.212.240 (talk) 06:10, 30 September 2009 (UTC)

 * The a is there because the definition of geometric progression does not assume that it starts with 1. It's probably simpler to just keep the a than to keep having to explain why it was taken out.--RDBury (talk) 13:20, 1 October 2009 (UTC)

 * The article deals with geometric series in general. I cannot see any good reason for giving anything less than the most general formulas. It is true that it is easy to derive the general formulas from more specific ones, but why is that a justification for not giving the formulas in their most general form? JamesBWatson (talk) 18:49, 4 October 2009 (UTC)

 * In addition to the above math-based reasons to retain the a, here's a encylcopedia-based suggestion: discussion of these series is taught in high school with the constant a (sometimes at least).  So retaining the a makes the page more useful to readers with that level of experience.  Citations for the assertion about math teaching include the "Kaplan SAT Subject Test Mathematics Level 2" 2010(?) edition.  The constant was included in my daughter's high school classes, although that hardly qualifies as a decent citation. DavidHolmes0 (talk) 21:35, 6 September 2010 (UTC)

Proof that 1 + 2 + 4 + 8 + ... = -1
n = 1 + 2 + 4 + 8 + 16 + ...

2n = 2 + 4 + 8 + 16 + 32 + ...

n = 1 + 2n

0 = 1 + n

n = -1

I talked to a mathematician about this proof and he didn't see any flaw in it. He said that the only thing that it proves is that if the series does converge, that it would converge to -1. An example of where it would converge to -1 would be in 2-adic. —The preceding unsigned comment was added by 61.108.11.194 (talk • contribs) 06:41, 8 February 2007 (UTC)

 * That sounds correct, and it could be made more precise. For example, you could say that any summation method for divergent series that is both stable and linear, and sums 1 + 2 + 4 + 8 + · · ·, sums it to −1. Melchoir 22:17, 8 February 2007 (UTC)

 * A work colleague gave me a variant of this proof, and referred to it as “proof that the universe is two's complement”. (The two's complement representation of −1 is all-bits-one, which in some sense ought to mean 1+2+4+⋯.)  Pjrm 06:28, 19 April 2007 (UTC)

Using the formula for the limiting sum anyway: = 1/(1-2) = -1

Looking at the series to the left of the term.. 1/2, 1/4.. etc we get limiting sum 1.

It's analogous to "running the integral" backwards of this exponential function and finding it's limiting area.

(remark)
 * I'd say its not 'n = 1 + 2n' but rather 'n = 1 + 2n-inf' —The preceding unsigned comment was added by Morgenrodeo (talk • contribs) 08:14, 28 March 2007 (UTC).

There is an error with the proof. you cannot multiply infinity by two since by definition it is the largest number possible. —The preceding unsigned comment was added by 172.207.30.115 (talk • contribs) 06:55, April 1, 2007 (UTC).

 * This isn't true. Infinity is not the "biggest number possible", it's not even a number. There are different types of infinity. For example the integers are infinite in number, and so are the real numbers, but there are far more real numbers than integers. The integers are countably infinite, the real numbers are uncountably infinite. The problem in the proof is that 1 + 2 + 4 + 8 + … is not well defined: it's infinite! It just doesn't make any sense to write 1 + 2 + 4 + 8 + … and expect it to mean anything. And so multiplying by 2 doesn't make any sense either. The formula for a geometric progression only works when | r | < 1. When | r | ≥ 1 then the sum diverges. The problem here is a common one; people are trying to apply a result without enforcing all of the prerequisite assumptions.  Dr Dec  ( Talk )    22:48, 30 September 2009 (UTC)

 * Absolutely. The above "proof" multiplies one infinite quantity by two, and then subtracts another infinite quantity from it... and then expects a meaningful, finite answer.  Most useful definitions of "infinity" in mathematics preclude that possibility.
 * Similar errors of subtracting infinite quantities lead to other nonsensical results. For example, here is a "proof" that 0=1, and that 0=2 also:
 * n = 1 + 1 + 1 + 1 + 1 + ...
 * n + 1 = 1 + 1 + 1 + 1 + 1 + ...
 * Thus n=n+1
 * Subtract n from both sides
 * Thus 0 = 1
 * n = 2 + 2 + 2 + 2 + 2 + ...
 * n + 2 = 2 + 2 + 2 + 2 + 2 + ...
 * Thus n=n+2
 * Thus 0 = 2
 * This mathematical fallacy is very much related to Hilbert's paradox of the Grand Hotel. Moxfyre (ǝɹʎℲxoɯ | contrib) 01:35, 2 October 2009 (UTC)
 * This mathematical fallacy is very much related to Hilbert's paradox of the Grand Hotel. Moxfyre (ǝɹʎℲxoɯ | contrib) 01:35, 2 October 2009 (UTC)
 * This mathematical fallacy is very much related to Hilbert's paradox of the Grand Hotel. Moxfyre (ǝɹʎℲxoɯ | contrib) 01:35, 2 October 2009 (UTC)

As presented in the original post this is not a fallacy; it is a perfectly valid proof that, as stated, if the series converges then it converges to -1. Since it is perfectly clear that it does not converge this result is of no conceivable value, but that is not at all the same as saying that it is fallacious. However, the section heading is nonsense: it is not a proof that the series converges to -1, and, in light of the statement from the original poster of what it proves, it is a bizarre heading. JamesBWatson (talk) 18:36, 4 October 2009 (UTC)

Partial Sum
The line reading: Here n = 1+2+4+8+…….. 2n = 2+4+8+16+…………

Here we can easily say 2n>n

2n-n = (2+4+8+16+…………) – (1+2+4+8+……..) 2n-n = -1 n = 2n+1, which contradict 2n>n so n ≠ -1

Don't merge individual series into here
The beauty of summary style is that we can develop examples in separate articles, which include unique applications and background. All convergent geometric series are not identical, and they can't be covered in a single article. Melchoir 18:47, 16 March 2007 (UTC)
 * Most of the content of the individual articles is already here — all convergent geometric series are (pretty much) identical except for where they're referred to in the literature. 1/2 + 1/4 + ... -> Zeno and 0.999, 9/10 + 9/100 + ... to 0.999, 1/2 - 1/4 + 1/8 - ... to Numbers, 1/4 + 1/16 + ... to the previous series (if it is notable), etc. Perhaps there should be an article should be examples of geometric series, split out from the "geometric series" section, here, but the ones here don't deserve an article because no one could type it it exactly, and search probably wouldn't fit it either.  (The spaced "dots" rather than unspaced dots or an ellipsis is another problem.... (!)).  — Arthur Rubin |  (talk) 22:40, 16 March 2007 (UTC)
 * Well, the individual articles are mostly substubs created without my knowledge; that's why they currently lack unique content. You could make them into redirects, but then I'd just have to recreate them when I get around to it.
 * Anyway, this article has become an unreadable, tangled list of boring quasi-information. It's a concrete argument against the incremental improvement model. As a reader, if I followed a link to some series and it took me here, I'd feel betrayed. Melchoir 23:21, 16 March 2007 (UTC)

I am for the merge unless the article can be expanded into something unique enough to merit its own article--Cronholm144 22:55, 12 May 2007 (UTC)

incorrect description of final outcome in one case

 * 1) Positive, the terms will all be the same sign as the initial term.
 * 2) Negative, the terms will alternate between positive and negative.
 * 3) Greater than 1, there will be exponential growth towards positive infinity.

I think that the third line should read:

Greater than 1, there will be exponential growth towards infinity (of the same sign as the first term).

eg, -1, -2, -4, -8 .... common ratio is 2, and is > 1

This is only the second 'post' I have made to Wikipedia. I would rather put this as a discussion item than wade in and edit the main page.

If someone with more experience agrees, would they please alter the main page.

Thank you Jsryork 11:21, 7 May 2007 (UTC)

What's the purpose of "a" in all these formulae?
Why is a needed in all these formulae? It's clear that it's a constant and therefore it can always come out of the summation... 83.67.217.254 20:41, 21 August 2007 (UTC)

 * It can come out of the summation when you're solving for the sum of the series, but it's needed when expressing the series as "a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number". If you didn't have a factor of a in each term, you could only express geometric series for which the first term is 1. —David Eppstein 05:09, 27 August 2007 (UTC)

 * That is not a property of the series. Its an application not an algebraic property. This is clutter. — Preceding unsigned comment added by 95.176.83.179 (talk) 10:37, 29 May 2011 (UTC)
 * Although not a significant improvement, it does allow the sum to be determined for an arbitrary geometric progression, not just one starting with "1", per David Eppstein's comment above. — Arthur Rubin  (talk) 14:43, 29 May 2011 (UTC)

Discrepancies in term numbering in formulae
1) This page has several discrepancies in numbering of terms:

"Elementary properties" says a_n = a *r^(n-1) i.e. terms start from n=1 however the summation formula and everything from the section "Geometric series" onwards uses: a_k = ar^k starting with k=0

2) The term-index used varies between n, k and even i

Please use one consistent notation and one term-index throughout; most people would favor the notation a_k = ar^k (i.e. numbering always starts from 0) and index k for simplicity. It also keeps the summation formula simpler.

Smcinerney 04:08, 3 September 2007 (UTC)

 * I have revised the Elementary Properties section to be consistent with the Series section. Journeyman (talk) 23:18, 26 March 2008 (UTC)

Geometric series article
Hi, I just added an article entirely on the subject of infinite geometric series. Geometric progressions and geometric series are important enough that they deserve separate articles. Jim 21:06, 22 September 2007 (UTC)

Formula for infinite series

 * $$\frac{1}{\frac{1}{a^1}+\frac{1}{a^2}+\frac{1}{a^3}+\frac{1}{a^4}+\frac{1}{a^5}\ldots}=\frac{1}{\sum_{n=1}^{\infty}\frac{1}{a^n}}=a-1;a>0$$

User:twentythreethousand 22:39, 27 September 2007

Infinite Series Explained
I had an assignment involving infinite geometric series and at the time I couldn't understand the justification for the formula presented here, as it looked like half of the formula just dissapeared. Later I found that it was because the r^infinity equalled zero. I tried to simplify this line of math so that others wouldn't be as confused as I was... is it mathematically correct tho, ne comments Laubpatr (talk) 04:34, 16 March 2008 (UTC)

 * The argument you give is not quite correct, since r∞ is not well-defined. However, it is true that if |r| < 1, then the value of rn can be made arbitrarily close to zero by selecting a sufficiently large value of n, that is, limn→∞ rn = 0. Michael Slone (talk) 08:23, 16 March 2008 (UTC)

derivation and simple example
The formula ar^(n-1) is easily derived and simplifies understanding. As my edits are always deleted by some insane, wide eyed, fool of a moderater I sha'n't waste my time producing one. But a derivation would be a worthy addition to the page.

Also a simple example would make this more accessable. Perhaps using the forumula to calculate compound interest.

Also, where are the link to the exponential function and the derivation of e?

Page needs a lot of work. —Preceding unsigned comment added by 91.111.137.74 (talk) 13:49, 16 August 2009 (UTC)

Error?
There appears to be an error in the geometric series section. The "convenient formula" appears to be wrong. —Preceding unsigned comment added by 67.230.157.187 (talk) 04:26, 1 October 2009 (UTC)

 * No,the formula is correct. We can derive it here. Let's ignore the factor of a, we could just multiply through by a at the and if we like. If Sn = 1 + r + r2 + … + rn then rSn = r + r2 + … + rn+1 and so (1 − r)S = 1 − rn+1. Dividing through by 1 − r gives
 * $$S_n = \frac{1-r^{n+1}}{1-r} = \frac{r^{n+1}-1}{r-1} \ . $$
 * Maybe the sign change confused you? In the article the numerator and denominator have both been multiplied by −1. But since (−1) ÷ (−1) = +1 there is no change in the answer.  Dr Dec  ( Talk )    10:34, 1 October 2009 (UTC)

P.S. When you post onto a talk page, please put your post at the bottom of the page. See the talk page article for more information. Dr Dec ( Talk )    10:34, 1 October 2009 (UTC)

start value? whazzat?
The introduction of the article currently says that the ratio between adjacent terms is "equal to the sequence's start value". That doesn't make sense to me, since I don't know what is meant by progression's start value. Perhaps the use of that phrase is a remnant from an earlier edition of the page wherein the constant "a" was not used?

In any case, can someone can clarify the phrase in the article, or remove it?

I know I'm supposed to Be Bold in editing, so maybe I should have removed the phrase, but this would have been my first edit to an article, and leading from ignorance was unappealing. Helpful suggestions are welcome. DavidHolmes0 (talk) 21:52, 6 September 2010 (UTC)
 * Please ignore my comments above, which, on later reading, don't seem to match the history of the article. What was I thinking?  DavidHolmes0 (talk) 15:02, 21 September 2010 (UTC)

Calculus on Geometric Series
Under the Geometric Series heading there is a line that says, "Differentiating this formula with respect to r allows us to arrive at formulae for sums of the form." This is a tad bit confusing because it is first unclear which formula "this formula" is. Apart from simply editing for clarity, should this material be moved to a section about calculus? Thelema418 (talk) 05:25, 23 August 2012 (UTC)

Merger proposal
I propose that Geometric progression be merged in part or whole into Geometric series. The concept of the progression or sequence is necessary to understand the series, so it is necessary in the Series article. For some reason, the series article concerns infinite series, but finite series are described in the geometric progression article. This is confusing because it makes it sound as though Geometric series refers to the infinite progression, and never a finite sequence. Thelema418 (talk) 02:20, 24 August 2012 (UTC)

 * I would agree; they should be together as are Arithmetic Progression and Arithmetic Series. It's confusing to have Geometric Progression and Geometric Series as separate, lengthy articles. Startswithj (talk) 18:48, 30 November 2013 (UTC)

 * I disagree with the merger proposal. See my comments on Talk:Geometric series. Jim.belk (talk) 05:02, 26 January 2014 (UTC)

 * I also disagree. A progression and a serie are totally differents objects. As far as the merged proposal is concerned: both progression and series are infinite, which do not forbid to look at finite inital sub-series. 2A02:1205:C6B7:F9C0:8150:27F3:EB3E:54DC (talk) 17:19, 22 July 2014 (UTC)

 * Agree. See my comments in Talk:Geometric series. Mario Castelán Castro (talk) 00:52, 8 February 2015 (UTC).

 * Oppose merge. As the IPV6 editor said, series (adding things up) and progressions (listing things one after the other in a certain order) are two very different things, and blurring the distinctions between the two does no favors to our readers. Additionally, the geometric series article is already on the long side, and if the two are merged it will become more difficult to find information about geometric progressions within them. I do think that the "series" part of the progression article is too long — it should be cut back to a brief overview of the main results for geometric series with a pointer to the other article. —David Eppstein (talk) 02:12, 8 February 2015 (UTC)
 * Oppose merge, Support two articles: I suggest that the articles Geometric progression and Geometric series be two different articles, as even the finite series stray from the topic. I would structure it as if something is related to the progression, it will be put into the progression, and if something is related to the series, it will be placed into the series. DS Crowned (talk)  06:32, 13 July 2016 (UTC)

Finding R
Is it possible to find R given the sum, first term, and number of terms? I've asked my math teachers this and they all said no. I'm assuming that the responses here will be the same, but even if it's not possible, how close can I get to finding the answer? — Preceding unsigned comment added by 71.206.163.33 (talk) 21:22, 25 August 2013 (UTC)
 * In a sense, yes, if r is positive. S(a,r,n) is an increasing function of r for r nonnegative, so there is a unique solution of S(a,r,n)=S given a, n, and S if S > a.  However, there isn't a "closed form" solution.  — Arthur Rubin  (talk) 01:03, 27 November 2013 (UTC)

An error in product section
I notice that the product equal to P=(a0*a1)^((n+1)/2) The correct formula should be P=(a0*a1)^(n/2)

 * No, the formula given, $$P = (a_0 a_n)^\frac{n+1}{2}$$, is correct. There are n + 1 terms since the first term is indexed by "0", and the proof is straight-forward. Bill Cherowitzo (talk) 18:48, 24 April 2015 (UTC)

dubious restrictions on a and r
Epp's discrete math book and Rosen's discrete math book place no restrictions on a and r. Since the sequences are defined using multiplication the restrictions are unnecessary. Occam's razor dictates that the restrictions be omitted. 68.81.2.180 (talk) 18:37, 25 May 2022 (UTC)

Problem with product formula
The given formula for the product of the first n elements obviously does not hold if a<0: For example, with a=-1, n=0 and r=1, the product formula yields -1 = 1. I cannot find any hint on additional constraints for a in the article, though. 2A02:908:3220:960:7488:8916:ED75:DE65 (talk) 10:55, 22 January 2024 (UTC)

title: Talk:Geometric progression


text:
List of wars involving Sweden

This is a list of wars involving the Kingdom of Sweden. There are legendary accounts of Swedish kings well into prehistory and they are mentioned by Tacitus in his Germania, but Olof Skötkonung (995–1022) was the first ruler documented to have been accepted by both the Swedes around Lake Mälaren and by the Geats around Lake Vättern. The modern state of Sweden considers itself to have been established on 6 June 1523 by the acclamation of Gustav Vasa as king which finally ended the Kalmar Union with Denmark, although the current Swedish constitution dates to 1974.

Military engagements since 1814 have not been formally declared wars.

title: List of wars involving Sweden


text:
Alessandro Cagliostro

Giuseppe Balsamo (in French usually Joseph Balsamo; 2 June 1743 – 26 August 1795), known by the alias Count Alessandro di Cagliostro, was an Italian occultist.

Cagliostro was an Italian adventurer and self-styled magician. He became a glamorous figure associated with the royal courts of Europe where he pursued various occult arts, including psychic healing, alchemy, and scrying. His reputation lingered for many decades after his death but continued to deteriorate, as he came to be regarded as a charlatan and impostor, this view fortified by the savage attack of Thomas Carlyle (1795–1881) in 1833, who pronounced him the "Quack of Quacks". Later works—such as that of W. R. H. Trowbridge (1866–1938) in his Cagliostro: the Splendour and Misery of a Master of Magic (1910), attempted a rehabilitation.

Origin
The history of Cagliostro is shrouded in rumour, propaganda, and mysticism. Some effort was expended to ascertain his true identity when he was arrested because of possible participation in the Affair of the Diamond Necklace. Johann Wolfgang von Goethe relates in his Italian Journey that the identification of Cagliostro with Giuseppe Balsamo was ascertained by a lawyer from Palermo who, upon official request, had sent a dossier with copies of the pertinent documents to France. Goethe met the lawyer in April 1787 and saw the documents and Balsamo's pedigree: Balsamo's great-grandfather Matteo Martello had two daughters: Maria, who married Giuseppe Bracconeri; and Vincenza, who married Giuseppe Cagliostro. Maria and Giuseppe Bracconeri had three children: Matteo; Antonia; and Felicità, who married Pietro Balsamo (the son of a bookseller, Antonino Balsamo, who had declared bankruptcy before dying at age 44). The son of Felicità and Pietro Balsamo was Giuseppe, who was christened with the name of his great-uncle and eventually adopted his surname, too. Felicità Balsamo was still alive in Palermo at the time of Goethe's travels in Italy, and he visited her and her daughter. Goethe wrote that Cagliostro was of Jewish origin, and it may be that the name "Balsamo" comes from the hebrew Baal Shem (Cagliostro himself publicly asserted that he was a disciple of Haĩm Falk, the Baal Shem of London).

Cagliostro himself stated during the trial following the Affair of the Diamond Necklace that he had been born of Christians of noble birth but abandoned as an orphan upon the island of Malta. He claimed to have travelled as a child to Medina, Mecca, and Cairo and upon return to Malta to have been admitted to the Sovereign Military Order of Malta, with whom he studied alchemy, the Kabbalah, and magic.

Early life
Giuseppe Balsamo was born to a poor family in Albergheria, which was once the old Jewish Quarter of Palermo, Sicily. Despite his family's precarious financial situation, his grandfather and uncles made sure the young Giuseppe received a solid education: he was taught by a tutor and later became a novice in the Catholic Order of St. John of God, from which he was eventually expelled. During his period as a novice in the order, Balsamo learned chemistry as well as a series of spiritual rites. In 1764, when he was twenty-one, he convinced Vincenzo Marano—a wealthy goldsmith—of the existence of a hidden treasure buried several hundred years previously at Mount Pellegrino. The young man's knowledge of the occult, Marano reasoned, would be valuable in preventing the duo from being attacked by magical creatures guarding the treasure. In preparation for the expedition to Mount Pellegrino, however, Balsamo requested seventy pieces of silver from Marano.

When the time came for the two to dig up the supposed treasure, Balsamo attacked Marano, who was left bleeding and wondering what had happened to the boy—in his mind, the beating he had been subjected to had been the work of djinns. The next day, Marano paid a visit to Balsamo's house in via Perciata (since then renamed via Conte di Cagliostro), where he learned the young man had left the city. Balsamo (accompanied by two accomplices) had fled to the city of Messina. By 1765–66, Balsamo found himself on the island of Malta, where he became an auxiliary (donato) for the Sovereign Military Order of Malta and a skilled pharmacist.

Travels
In early 1768 Balsamo left for Rome, where he managed to land himself a job as a secretary to Cardinal Orsini. The job proved boring to Balsamo and he soon started leading a double life, selling magical "Egyptian" amulets and engravings pasted on boards and painted over to look like paintings. Of the many Sicilian expatriates and ex-convicts he met during this period, one introduced him to a fourteen-year-old girl named Lorenza Seraphina Feliciani (ca. 8 April 1751 – 1794), known as Serafina, whom he married 1768.

The couple moved in with Lorenza's parents and her brother in the vicolo delle Cripte, adjacent to the strada dei Pellegrini. Balsamo's coarse language and the way he incited Lorenza to display her body contrasted deeply with her parents' deep-rooted religious beliefs. After a heated discussion, the young couple left. At this point, Balsamo befriended Agliata, a forger and swindler, who proposed to teach Balsamo how to forge letters, diplomas and myriad other official documents. In return, Agliata sought sexual intercourse with Balsamo's young wife, a request to which Balsamo acquiesced.

The couple traveled together to London, where Balsamo, now styling himself with one of several pseudonyms and self-conferred titles before settling on "Count Alessandro di Cagliostro", allegedly met the Comte de Saint-Germain. Cagliostro traveled throughout Europe, especially to Courland, Russia, Poland, Germany, and later France. His fame grew to the point that he was even recommended as a physician to Benjamin Franklin during a stay in Paris.

On 12 April 1777, "Joseph Cagliostro" was admitted as a Freemason of the Espérance Lodge No. 289 in Gerrard Street, Soho, London. In December 1777 Cagliostro and Serafina left London for the mainland, after which they travelled through various German states, visiting lodges of the Rite of Strict Observance looking for converts to Cagliostro's "Egyptian Freemasonry". In February 1779 Cagliostro traveled to Mitau, (nowadays Latvia), where he met the poet Elisa von der Recke. In September 1780, after failing in Saint Petersburg to win the patronage of Russian Tsaritsa Catherine the Great, the Cagliostros made their way to Strasbourg, at that time in France. In October 1784, the Cagliostros travelled to Lyon. On 24 December 1784 they founded the co-Masonic mother lodge La Sagesse Triomphante of his rite of Egyptian Freemasonry at Lyon. In January 1785 Cagliostro and his wife went to Paris in response to the entreaties of Cardinal Rohan.

Affair of the diamond necklace
Cagliostro was prosecuted in the Affair of the Diamond Necklace which involved Marie Antoinette and Cardinal Prince Louis de Rohan, and was held in the Bastille for nine months but finally acquitted, when no evidence could be found connecting him to the affair. Nonetheless, he was banished from France by order of Louis XVI, and departed for England. There he was accused by French expatriate Theveneau de Morande of being Giuseppe Balsamo, which he denied in his published Open Letter to the English People, forcing a retraction and apology from Morande.

Betrayal, imprisonment, and death
Cagliostro left England to visit Rome, where he met two people who proved to be spies of the Inquisition. Some accounts hold that his wife was the one who initially betrayed him to the Inquisition. On 27 December 1789, he was arrested for attempting to found a Masonic lodge in Rome, and was imprisoned in the Castel Sant'Angelo. He was tried and originally sentenced to death but the sentence was later commuted to life imprisonment at the Forte di San Leo, where he would die on 26 August 1795.

Legacy
Portuguese author Camilo Castelo Branco credits to Balsamo the creation of the Egyptian Rite of the Freemasons and intensive work in the diffusion of Freemasonry, by opening lodges all over Europe and by introducing the acceptance of women into the community. The idea of an "Egyptian freemasonry" was maintained in Italy by the Rite of Misraim, founded in 1813 by the three Jewish Bédarride brothers and in France, the Rite of Memphis founded in 1838 by Jacques Etienne Marconis de Nègre; these unified under Giuseppe Garibaldi as the Ancient and Primitive Rite of Memphis-Misraïm in 1881.

Cagliostro was an extraordinary forger. Giacomo Casanova, in his autobiography, narrated an encounter in which Cagliostro was able to forge a letter by Casanova, despite being unable to understand it. Occult historian Lewis Spence comments in his entry on Cagliostro that the swindler put his finagled wealth to good use by starting and funding a chain of maternity hospitals and orphanages around the continent. He carried an alchemistic manuscript The Most Holy Trinosophia amongst others with him on his ill-fated journey to Rome, and it is alleged that he wrote it. Occultist Aleister Crowley believed Cagliostro was one of his previous incarnations.

Drama

 * Catherine the Great wrote two skits lampooning Cagliostro in the guise of characters loosely based upon him.
 * Johann Wolfgang Goethe wrote a comedy based on Cagliostro's life, also in reference to the Affair of the Diamond Necklace, The Great Cophta (Der Groß-Coptha) which was published in 1791.
 * Latvian playwright Mārtiņš Zīverts wrote the play Kaļostro Vilcē (Cagliostro in Vilce) in 1967.

Literature

 * Alexandre Dumas, père used Cagliostro in several of his novels (especially in Joseph Balsamo and in Le Collier de la Reine where he claims to be over 3,000 years old and to have known Helen of Troy).
 * George Sand includes Cagliostro as a minor character in her historical novel, The Countess of Rudolstadt (1843).
 * Aleksey Nikolayevich Tolstoy wrote the supernatural love story Count Cagliostro, in which the Count brings to life a long dead Russian princess by materializing her from her portrait. The story was made into a 1984 Soviet TV movie Formula of Love.
 * Cagliostro is featured in three stories by Rafael Sabatini, namely "The Lord of Time", "The Death Mask" and "The Alchemical Egg", which are included in Sabatini's collection Turbulent Tales.
 * In "The Sandman" by ETA Hoffmann. Spalanzani is said to resemble a portrait of Cagliostro by Chodowiecki.
 * In "The Book and the Beast", a short story by Robert Arthur, Jr., a grimoire attributed to Caliostro causes the gruesome death of those foolish enough to examine it, until a fire destroys it.
 * He is mentioned in the novel Kun Lun by Kilburn Hall (2014), where it is revealed that Alessandro Cagliostro, Joseph and Giuseppe Balsamo are just a few of the names that the time traveler Count St. Germain has used throughout history.
 * Friedrich Schiller started, but not finish, the novel Der Geisterseher (The Ghost-Seer) between 1786 and 1789, concerning him.
 * Harry Stephen Keeler paid tribute to the magician in his novel The Spectacles of Mr. Cagliostro.
 * He is a character in Robert Anton Wilson's The Historical Illuminatus Chronicles.
 * He is frequently mentioned Umberto Eco's novel Foucault's Pendulum.
 * Mikhail Kuzmin wrote a novella called The Marvelous Life of Giuseppe Balsamo, Count Cagliostro (1916).
 * He is a character in Psychoshop, a novel by Alfred Bester and Roger Zelazny.
 * Josephine Balsamo, a descendant of Joseph Balsamo who calls herself Countess Cagliostro, appears in Maurice Leblanc's Arsene Lupin novels.
 * Cagliostro makes a number of appearances as a vampire in Kim Newman's Anno Dracula series of novels.
 * There are numerous references to Cagliostro in the detective novel He Who Whispers by John Dickson Carr (aka Carter Dickson), one of his Dr Gideon Fell mysteries, published by Hamish Hamilton (UK) & Harper (USA) in 1946. In this book, a French professor, Georges Antoine Rigaud, has written a history: Life of Cagliostro. An attempted murder committed in He Who Whispers is similar in technique to part of an initiation ceremony undergone by Cagliostro into the lodge of a secret society. Cagliostro Street appears as a location in Carr's 1935 novel The Hollow Man (published in the US as The Three Coffins).
 * He is a character in the 1997 novel 'Superstition' by David Ambrose. He is an acquaintance of the fictional Adam Wyatt.
 * He is often mentioned in the novel Napoleon's Pyramids by William Dietrich in connection with Freemasons and ancient Egyptian artifacts.
 * In Robert A. Heinlein's Glory Road, Star uses "Balsamo" as an alias, and refers to Giuseppe as her uncle.
 * William Bolitho Ryall's Twelve Against The Gods has a section on Cagliostro.

Comics

 * The Phantom comic book (based on a comic strip of the same name) featured Cagliostro as a character in the story "The Cagliostro Mystery" from 1988. written by Norman Worker and drawn by Carlos Cruz.
 * The third Kid Eternity comic book, published in 1946, featured Cagliostro's risen spirit.
 * In the DC Comics universe, Cagliostro is described as an immortal (JLA Annual 2), a descendant of Leonardo da Vinci as well as an ancestor of Zatara and Zatanna (Secret Origins 27).
 * In Marvel Comics' Tomb of Dracula and Dracula Lives comic books, Cagliostro is a frequent enemy of Dracula.
 * The manga Rozen Maiden gives Count Cagliostro as one of many aliases adopted by the legendary dollmaker Rozen. He was shown to be in prison whittling wood.
 * He is a character in Todd McFarlane's comic bookSpawn. He was introduced to the series by writer Neil Gaiman. Here, Cogliostro was once a spawn of Hell bound to his duty to the daemon Malebolgia. Having freed himself of the curse through alchemy and sorcery, he teaching Spawn to do the same throughout the series.

Video games

 * Cagliostro is a playable character in the Japanese Mobile game Granblue Fantasy.
 * Payday 2 by Overkill and Starbreeze studios features Cagliostro's manuscript as a key story item and opens a deep mystery within the game involving secret societies, immortality and nephilims.
 * Cagliostro is a villain in the Spiders video game Steelrising. His penchant for magic and alternative medicine is referenced; for example, in one scene, he is shown practicing hypnosis with a pendulum.
 * Cagliostro is featured in Fate/Grand Order as a Pretender-class servant.

Music

 * He appears as a principal character in the 1794 opera Le congrès des rois, a collaborative work of 12 composers.
 * The French composer Victor Dourlen (1780–1864) composed the first act to Cagliostro, ou Les illuminés which premiered on 27 November 1810. The second and third acts were composed by Anton Reicha (1770–1836).
 * The Irish composer William Michael Rooke (1794–1847) wrote an unperformed work Cagliostro.
 * Adolphe Adam wrote the opéra comique Cagliostro which premiered on 10 February 1844.
 * Albert Lortzing wrote in 1850 the libretto for a comic opera in three acts, Cagliostro, but did not compose any music for it.
 * Johann Strauß (Sohn) wrote the operetta Cagliostro in Wien (Cagliostro in Vienna) in 1875.
 * The French composer Claude Terrasse (1867–1923) wrote Le Cagliostro which premiered in 1904.
 * The Polish composer Jan Maklakiewicz (1899–1954) wrote the ballet in three scenes Cagliostro w Warszawie which premiered in 1938.
 * The Romanian composer Iancu Dumitrescu (1944–) wrote the 1975 work Le miroir de Cagliostro for choir, flute and percussion.
 * The American composer John Zorn (1953–) composed Cagliostro for solo viola in 2015. The performer uses two bows in the right hand to play on all four strings at once throughout the work.
 * The opera Cagliostro by the Italian composer Ildebrando Pizzetti (1880–1968) was performed on Italian radio in 1952 and at La Scala on 24 January 1953.
 * The comic opera Graf Cagliostro was written by Mikael Tariverdiev in 1983.

Film

 * Cagliostro has been portrayed in film by:
 * Fryderyk Jarossy (Kaliostro, 1918)
 * Reinhold Schünzel (The Count of Cagliostro, 1920)
 * Hans Stüwe (Cagliostro, 1929)
 * Ferdinand Marian (Münchhausen, 1943)
 * Orson Welles (Black Magic, 1949)
 * Howard Vernon (Erotic Rites of Frankenstein, 1972)
 * Jean Marais (, 1973, TV miniseries)
 * Bekim Fehmiu (Cagliostro, 1975)
 * Nodar Mgaloblishvili (Formula of Love, 1984, TV film)
 * Nicol Williamson (Spawn, 1997)
 * Christopher Walken (The Affair of the Necklace, 2001)
 * Robert Englund (The Return of Cagliostro, 2003)
 * In the 1943 German epic Münchhausen, Cagliostro appears as a powerful, morally ambiguous magician portrayed by Ferdinand Marian.
 * The French film director Georges Méliès (1861–1938) directed the 1899 film Le Miroir de Cagliostro.
 * The Japanese animated movie The Castle of Cagliostro draws on Maurice Leblanc's Arsène Lupin novels and has the gentleman thief's half-Japanese grandson as the protagonist. Lazare d'Cagliostro appears as the main antagonist of the film, a ruler of a fictional country who influences the world's economy through counterfeiting (inspired by the 1977 fanfiction The Justice of Arsène Lupin).
 * The Mummy (1932), starring Boris Karloff, was adapted from an original story treatment by Nina Wilcox Putnam titled "Cagliostro". Based on Cagliostro and set in San Francisco, the story was about a 3000-year-old magician who survives by injecting nitrates.
 * Cagliostro and his wife, Lorenza, appear as antagonists in the 2006 anime Le Chevalier d'Eon. While Cagliostro is mostly portrayed as a bumbling money-grubber, Lorenza is shown to have arcane magic powers.
 * In the Marvel Cinematic Universe, Cagliostro is a sorcerer, and is mentioned often in Doctor Strange (2016). The Book of Cagliostro: Study of Time is an ancient artifact containing several dark spells. The spin-off Disney+ series What If...? mentions him as one who could break an absolute point in time. Doctor Strange read the lost books of Cagliostro and reversed an absolute point in time, much like the books' author.

Television

 * He is a whimsical villainous alchemist character in the TV anime Senki Zesshou Symphogear AXZ.
 * He appears as a villainous magician in an episode of the 1960s series Thriller, entitled "The Prisoner in the Mirror"; he is played by Henry Daniell and Lloyd Bochner.
 * In "Diana's Disappearing Act", an 1978 episode of the Wonder Woman TV series, a descendant of Cagliostro's (played by Dick Gautier) is the villain. Attempting alchemy, he succeeds to the extent of turning lead into gold for a time, after which it reverts back to its original form. The long-lived Wonder Woman says that she faced his ancestor, the original count, in the past.
 * In Samurai Jack (the seventh episode of the third season), the title character follows a quest for the crystal of Cagliostro. This episode contains an homage to The Castle of Cagliostro by way of Jack receiving aid from a thief based directly on Daisuke Jigen.
 * The 2016 Lupin III yearly special featured a hunt for the treasure of Cagliostro. Prior to this, the name was also used for the 1979 Lupin III theatrical release The Castle of Cagliostro, though with little relation to the historical Cagliostro.
 * In The Twilight Zone (2002 TV series), Episode 36 "The Pharaoh's Curse", an up-and-coming illusionist strives to learn the secrets behind a centuries-old illusion, which has been purportedly handed down from magician masters Harry Houdini, Frederick Eugene Powell, and back originally to Cagliostro himself.

title: Alessandro Cagliostro