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Series whose partial sums eventually only have a fixed number of terms after cancellation
In mathematics, a telescoping series is a series whose general term t n {\displaystyle t_{n}} is of the form t n = a n − a n − 1 {\displaystyle t_{n}=a_{n}-a_{n-1}}
Telescoping_series
Technology in firearms
magazine located in the grip that do not use a telescoping bolt, such as the Kel-Tec SUB-2000. The telescoping bolt concept first appeared on semi-automatic
Telescoping_bolt
Property of an iterated binary operation
"Telescoping Series - Product". Brilliant Math & Science Wiki. Eric W. Weisstein. "Telescoping Sum". MathWorld. Po-Shen Loh (2003-06-18). "Telescoping
Telescoping_(mathematics)
Infinite sum
{\pi }{4}},} the Leibniz formula for π . {\displaystyle \pi .} A telescoping series ∑ n = 1 ∞ ( b n − b n + 1 ) {\displaystyle \sum _{n=1}^{\infty
Series_(mathematics)
Infinite series summing alternating 1 and -1 terms
sum of the series 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + … {\displaystyle 1-1+1-1+1-1+1-1+\ldots } would be to treat it like a telescoping series and perform
Grandi's_series
Number, product of consecutive integers
the reciprocals of the positive pronic numbers (excluding 0) is a telescoping series that sums to 1: ∑ i = 1 ∞ 1 i ( i + 1 ) = 1 2 + 1 6 + 1 12 + 1 20
Pronic_number
Canadian documentary series
Telescope is a Canadian documentary series which aired on CBC Television between 1963 and 1973. The series was hosted by Fletcher Markle, which profiled
Telescope_(TV_series)
Relationship between derivatives and integrals
integral. Mathematics portal Differentiation under the integral sign Telescoping series Fundamental theorem of calculus for line integrals Notation for differentiation
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Property of a mathematical operation
Light's associativity test Telescoping series, the use of addition associativity for cancelling terms in an infinite series A semigroup is a set with an
Associative_property
Topics referred to by the same term
recently Telescoping (mathematics), a general cancellation phenomenon in iterated expressions Telescoping series, in mathematics Telescoping generations
Telescope_(disambiguation)
Figurate number
\over {n^{2}+n}}=2.} This can be shown by using the basic sum of a telescoping series: ∑ n = 1 ∞ 1 n ( n + 1 ) = 1. {\displaystyle \sum _{n=1}^{\infty }{1
Triangular_number
numbers (products of two consecutive integers) (excluding 0) is 1 (see Telescoping series). The difference between H k ( n + 1 ) {\displaystyle H_{k(n+1)}}
List_of_sums_of_reciprocals
Branch of mathematics that studies dynamical systems
as N {\displaystyle N} grows, while for the latter part, from the telescoping series one would have: lim N → ∞ 1 N ∑ n = 0 N − 1 U n ( I − U ) = lim N
Ergodic_theory
Italian physicist and mathematician (1608–1647)
{\displaystyle a_{0},a_{1},a_{2},\ldots } and showed the corresponding telescoping series ( a 0 − a 1 ) + ( a 1 − a 2 ) + ⋯ {\displaystyle (a_{0}-a_{1})+(a_{1}-a_{2})+\cdots
Evangelista_Torricelli
Addition of several numbers or other values
{\displaystyle f(n)-f(m)=\sum _{i=m}^{n-1}(f(i+1)-f(i)).} This is known as a telescoping series and is the analogue of the fundamental theorem of calculus in calculus
Summation
Numeral system in combinatorics
subsequent terms cancel each other, leaving the first and last term (see Telescoping series). However, when using Arabic numerals to write the digits (and not
Factorial_number_system
NASA infrared space telescope launched in 2026
The Nancy Grace Roman Space Telescope is a NASA infrared space telescope that was launched on a trajectory toward a Sun–Earth L2 orbit on 30 August 2026
Nancy Grace Roman Space Telescope
Nancy_Grace_Roman_Space_Telescope
Difference between logarithm and harmonic series
that its partial sums form a telescoping series, such that the series again yields a limit between the harmonic series and a shifted natural logarithm
Euler's_constant
Creepshow Anthology
The following is a list of episodes of the American anthology television series Creepshow, which premiered on Shudder on September 26, 2019. As of October
List of Creepshow (TV series) episodes
List_of_Creepshow_(TV_series)_episodes
American sci-fi television series
fiction television series created by David Benioff, D. B. Weiss, and Alexander Woo. It is the third adaptation of the Chinese novel series Remembrance of
3_Body_Problem_(TV_series)
Mathematical method
{1}{6}}+{\frac {1}{6}}-{\frac {1}{10}}+\cdots \end{aligned}}} This is a telescoping series with sum value 1⁄2. In this case γ := lim n → ∞ 1 16 n 2 − 1 1 4 n
Kummer's transformation of series
Kummer's_transformation_of_series
Future observatory in the United States
The Thirty Meter Telescope (TMT) is a proposal for an extremely large telescope (ELT), intended to be built on Mauna Kea, on the island of Hawai'i. Scientists
Thirty_Meter_Telescope
Mathematical sequence satisfying a specific pattern
and using the technique of telescoping series (second equality) and the formula for the sum of a finite geometric series (fifth equality) gives ( 1 −
Arithmetico-geometric sequence
Arithmetico-geometric_sequence
The history of the telescope can be traced to before the invention of the earliest known telescope, which appeared in 1608 in the Netherlands, when a
History_of_the_telescope
Method of analyzing the amortized complexity of a data structure
(S_{0})),} where the sequence of potential function values forms a telescoping series in which all terms other than the initial and final potential function
Potential_method
Polyhedral number representing a tetrahedron
tetrahedral numbers' reciprocals is 3/2, which can be derived using telescoping series: ∑ n = 1 ∞ 6 n ( n + 1 ) ( n + 2 ) = 3 2 . {\displaystyle \sum _{n=1}^{\infty
Tetrahedral_number
NASA/ESA/CSA space infrared telescope launched in 2021
Webb's orientation. On 31 December 2021, the ground team extended the two telescoping "mid booms" from the left and right sides of the observatory. The left
James_Webb_Space_Telescope
Criterion for the convergence of a series
\infty }\zeta _{n}a_{n}=L} exists. This implies that the positive telescoping series ∑ n = 1 ∞ ( ζ n a n − ζ n + 1 a n + 1 ) {\displaystyle \sum _{n=1}^{\infty
Ratio_test
Number in the 5th cell of any row of Pascal's triangle
reciprocals of all pentatope numbers is 4/3. This can be derived using telescoping series. ∑ n = 1 ∞ 4 ! n ( n + 1 ) ( n + 2 ) ( n + 3 ) = 4 3 . {\displaystyle
Pentatope_number
Method for solving differential equations
^{3}}{3}})x+({\frac {-2\pi ^{3}}{3}}+{\frac {2\pi ^{6}}{9}})x+...} Since this is a telescoping series, we can see that every terms after c o s ( x ) {\displaystyle cos(x)}
Adomian_decomposition_method
Documentary TV series and book by James Burke
science education television series created, written, and presented by British science historian James Burke. The series was produced and directed by
Connections (British TV series)
Connections_(British_TV_series)
Type of optical telescope
A refracting telescope (also called a refractor or dioptric telescope) is a type of optical telescope that uses a lens as its objective to form an image
Refracting_telescope
Overview of and topical guide to combinatorics
Mathematical induction Recurrence relation, telescoping series Generating functions as an application of formal power series Cyclic sieving Schrödinger method Exponential
Outline_of_combinatorics
preschool animated children's television series based on the characters from Mick Inkpen's Kipper the Dog picture book series which ran from 5 September 1997 to
List_of_Kipper_episodes
Manufacturer of telescopes, etc., based in Watsonville, California
telescopes under their ETX series (Everybody's Telescope). They were first produced in 90 mm (3-1/2") Maksutov Cassegrain telescope in 1996. They range in
Meade_Instruments
German television series
Das Boot is a German television series produced by Bavaria Fiction for Sky One and a sequel to the film of the same name, based on Lothar-Günther Buchheim's
Das_Boot_(2018_TV_series)
Historical drama television series (2005–2007)
series ran for two seasons out of the planned five due to high production cost; much of the material for the third and fourth seasons was telescoped into
Rome_(TV_series)
series – see Taylor series Binomial series – the Maclaurin series of the function f given by f(x) = (1 + x) α Telescoping series Alternating series Geometric
List_of_real_analysis_topics
1992 Canadian children's TV series
adventure fantasy television series for children, originally broadcast from 1992 to 1994 on CBC Television. In the series pilot, 11-year-old Jay (Illya
The_Odyssey_(TV_series)
NASA/ESA space telescope launched in 1990
The Hubble Space Telescope (HST or Hubble) is a space telescope that was launched into low Earth orbit in 1990 and remains in operation. It was not the
Hubble_Space_Telescope
2015 American TV series or program
PBS series, Breakthrough: The Ideas That Changed the World, hosted by Patrick Stewart began airing in 2019 with six episodes, covering "The Telescope",
Breakthrough_(TV_series)
Children's live-action educational television series
from Balamory which had the same similarities in concept as Me Too!. The series follows six parents and their children, including Rebecca, Jack, Kai and
Me_Too!_(TV_series)
superhero series Heroes (2006–2010), Kirby Reed in the slasher horror franchise Scream (2011–2023), and Juliette Barnes in the ABC/CMT musical drama series Nashville
Hayden Panettiere filmography and discography
Hayden_Panettiere_filmography_and_discography
Telescope in the Atacama Desert, Chile
The Very Large Telescope (VLT) is an astronomical facility operated since 1998 by the European Southern Observatory, located on Cerro Paranal in the Atacama
Very_Large_Telescope
Algorithm for computing greatest common divisors
time for all of the steps of the algorithm can be analyzed using a telescoping series, showing that it is also O(h2). Modern algorithmic techniques based
Euclidean_algorithm
Short film series
Mickey Mouse Cartoons) is a series of American animated comedy short films produced by Walt Disney Productions. The series started in 1928 with Steamboat
Mickey_Mouse_(film_series)
Documentary strand presented by the BBC
part of BBC Film since 2020. Episodes from the 2015–2016 series Episodes from the 2014–2015 series: 24. Shooting Bigfoot: America's Monster Hunters 8. Smash
Storyville_(TV_series)
1964 Australian TV series or program
The series was made with cooperation from the CSIRO with pivotal scenes shot on location at the Parkes Observatory and the facility's radio telescope featured
The Stranger (Australian TV series)
The_Stranger_(Australian_TV_series)
Polish satirical comedy television series
satirical comedy television series. It premiered on Netflix on 13 December 2023. The third season premiered on 5 August 2026. The series follows the pursuits
1670_(TV_series)
2015 British television series
And Then There Were None is a 2015 mystery thriller television series that was first broadcast on BBC One from 26 to 28 December 2015. The three-part programme
And Then There Were None (TV series)
And_Then_There_Were_None_(TV_series)
Type of telescope mount and related software
astronomy, "GoTo" refers to a type of telescope mount and related software that can automatically point a telescope at astronomical objects that the user
GoTo_(telescopes)
Activity of building telescopes as a hobby
taking a series of measurements with the tester. This data is then reduced and graphed against an ideal parabolic curve. Some amateur telescope makers use
Amateur_telescope_making
American retail company
assemblies or "OTA") a variety of mounts. Orion also sold a series of Dobsonian telescopes that come in "Classic" and "IntelliScope" versions, the latter
Orion_Telescopes_&_Binoculars
United States historic place
cold winters, and the conical roof accommodated the gasholder, a telescoping series of cylinders that rose within guides as the volume of gas increased
Saratoga Gas, Electric Light and Power Company Complex
Saratoga_Gas,_Electric_Light_and_Power_Company_Complex
Topics referred to by the same term
mathematical concept Collapsing manifold Collapse, the action of collapsing or telescoping objects Collapsing user interface elements Accordion (GUI) -- collapsing
Collapse
Radio telescope in Bad Münstereifel, Germany
The Effelsberg 100-metre Radio Telescope is a radio telescope in the Ahr Hills (part of the Eifel) in Bad Münstereifel, Germany. Inaugurated in 1972, for
Effelsberg 100-metre Radio Telescope
Effelsberg_100-metre_Radio_Telescope
French film
The Arthur series refers to a series of fantasy novels for children written by Luc Besson, a film director and producer, and published from 2002 to 2005
Arthur_(Besson_book_series)
Type of motorcycle
Honda replaced the Super-Cub-style cantilever-toggle front end with telescoping-tube similar to those found on most motorcycles. This increased travel
Honda_CT90
Historic house in Maine, United States
large 2+1⁄2-story main block connected to a large carriage barn via a telescoping series of ells. The main block is oriented facing south, with a side gable
J.L._Prescott_House
Historic house in Maine, United States
fan. A hip-roofed porch extends across the southern facade, and a telescoping series of ells extend from the rear of the main block to join it to a large
Benjamin_Cleaves_House
Canadian television director (1936–2026)
flagship series The Way It Is, and the ACTRA award-winning three-part documentary series Centennial. He directed profiles for CBC's Telescope series of Oscar
Perry_Rosemond
Radio telescope in Spain
The IRAM 30m telescope is a radio telescope, located in the Sierra Nevada Mountain Range, Spain. It is operated by the Institute for Radio Astronomy in
IRAM_30m_telescope
Global radio telescope array
The Event Horizon Telescope (EHT) is a telescope array consisting of a global network of radio telescopes. The EHT project combines data from several
Event_Horizon_Telescope
British stop-motion animated television series (1984–1990)
The Wind in the Willows is a British stop motion animated television series that was originally broadcast between 1984 and 1987, based on characters from
The Wind in the Willows (TV series)
The_Wind_in_the_Willows_(TV_series)
American action television series (1966–1967)
the performers' faces. In a technological update, the Hornet carried a telescoping device called the Hornet's Sting, which projected ultrasonic soundwaves
The_Green_Hornet_(TV_series)
Space observatory
Project was the final mission in the High Energy Astronomy Observatory series, which was managed by the NASA Marshall Space Flight Center (MSFC), where
High Energy Astronomy Observatory 3
High_Energy_Astronomy_Observatory_3
X-ray source magnifier
A Wolter telescope is a telescope for X-rays that only uses grazing incidence optics – mirrors that reflect X-rays at very shallow angles in the regime
Wolter_telescope
NASA infrared space telescope (2003–2020)
The Spitzer Space Telescope, formerly the Space Infrared Telescope Facility (SIRTF), was an infrared space telescope that was active between 2003 and 2020
Spitzer_Space_Telescope
Optical telescope dedicated to observing the Sun
The Dunn Solar Telescope, also known as the Richard B. Dunn Solar Telescope, is a unique vertical-axis solar telescope that specializes in high-resolution
Richard B. Dunn Solar Telescope
Richard_B._Dunn_Solar_Telescope
Astronomical observatory in Los Angeles County, California, US
contains two historically important telescopes: the 100-inch (2.5 m) Hooker telescope, which was the largest aperture telescope in the world from its completion
Mount_Wilson_Observatory
British TV sitcom
From the second series, Sally Bretton replaced Dodds as Lucy, Lee's new landlady and Tim's sister. At the end of the seventh series, Lee proposes to
Not_Going_Out
American-Canadian Abenaki artist and filmmaker
interviewed about her film about the issue in a report for CBC-TV's Telescope series, which was seen by NFB producers Joe Koenig and Bob Verrall. They invited
Alanis_Obomsawin
Radio telescope in Green Bank, WV, US
Robert C. Byrd Green Bank Telescope (GBT) in Green Bank, West Virginia, US is the world's largest fully steerable radio telescope, surpassing the Effelsberg
Green_Bank_Telescope
1986 American animated television series
(previously called Ghostbusters) is a 1986 American animated television series created by Filmation and distributed by Tribune Entertainment, serving as
Ghostbusters_(1986_TV_series)
Radio telescope at Jodrell Bank Observatory, Cheshire, England
The Lovell Telescope (/ˈlʌvəl/ LUV-əl) is a radio telescope at Jodrell Bank Observatory, near Goostrey, Cheshire, in the north-west of England. Completed
Lovell_Telescope
Chinese telescope
study of the Wide Field Survey Telescope (WFST)". Society of Photo-Optical Instrumentation Engineers (Spie) Conference Series. Advanced Optical Design and
Wide_Field_Survey_Telescope
American television sitcom (1998–1999)
their first television series since Full House ended in 1995. The show aired from September 25, 1998, to July 9, 1999. The series was produced by Griffard/Adler
Two of a Kind (American TV series)
Two_of_a_Kind_(American_TV_series)
Alvin and the Chipmunks is an American animated television series featuring the Chipmunks, produced by Bagdasarian Productions in association with Ruby-Spears
List of Alvin and the Chipmunks (1983 TV series) episodes
List_of_Alvin_and_the_Chipmunks_(1983_TV_series)_episodes
Gamma ray detector
the Imaging Gamma-Ray Telescope COMPTEL aboard the Compton Gamma-Ray Observatory". The Astrophysical Journal Supplement Series. 86: 657. doi:10.1086/191794
Compton_telescope
American TV series
In Search of... is an American television series that was broadcast weekly from 1976 to 1982, devoted to mysterious phenomena. It was created after the
In_Search_of..._(TV_series)
Ability of any image-forming device to distinguish small details of an object
describes the ability of any image-forming device such as an optical or radio telescope, a microscope, a camera, or an eye, to distinguish small details of an
Angular_resolution
American animated series
The Mummy (also known as The Mummy: The Animated Series) is an American animated series produced by Universal Cartoon Studios based on the 1999 film of
The_Mummy_(TV_series)
Extending boom design
fully extended length depending on the number of stages. Some pneumatic telescoping units are manufactured with retracted lengths of under 15% of overall
Telescopic_cylinder
Criterion for convergence of continued fractions
{1}{\left|B_{n}\right|}}} Since the right-hand side forms a telescoping series, it is easy to see that | a 1 | | B 0 B 1 | + | a 1 a 2 | | B 1 B
Śleszyński–Pringsheim_theorem
Array of robotic optical telescopes
signals. The array consists of a network of telescope systems, with each system consisting of eight 0.4m telescopes on a single mounting. As of May 2023 the
GOTO_(telescope_array)
Works in Oxford University Press series
Very Short Introductions is a series of books published by Oxford University Press. Greer, Shakespeare: ISBN 978-0-19-280249-1. Wells, William Shakespeare:
List of Very Short Introductions books
List_of_Very_Short_Introductions_books
Inspector Montalbano is an Italian police procedural television series. Since 6 May 1999, 37 original episodes have been produced and broadcast by RAI
List of Inspector Montalbano episodes
List_of_Inspector_Montalbano_episodes
Children's television series
children's television series that was developed and produced by Patricia Edgar and broadcast on ABC Television from 1992 until the series ended in 1995. Each
Lift Off (Australian TV series)
Lift_Off_(Australian_TV_series)
Elevated photography is the process of taking aerial photos using a telescoping pole or mast, or other aerial or elevated support systems, to emulate
Elevated_photography
2016 American science fiction television series
Colony is an American science fiction drama television series created by Carlton Cuse and Ryan J. Condal, starring Josh Holloway and Sarah Wayne Callies
Colony_(TV_series)
British documentary TV series on Western art and philosophy (BBC 1969)
Personal View by Kenneth Clark is a 1969 British television documentary series written and presented by the art historian Sir Kenneth Clark. Its thirteen
Civilisation_(TV_series)
K. "The Atacama Submillimeter Telescope Experiment". The Cool Universe: Observing Cosmic Dawn, ASP Conference Series, Vol. 344, Proceedings of the conference
List_of_radio_telescopes
Czechoslovak submachine gun
trigger is released or the magazine is empty. The Sa 23 series were submachine guns with a telescoping bolt, in which the forward part of the moving bolt extends
Sa_23
Telescope at the South Pole
South Pole Telescope (SPT) is a 10-metre (390 in) diameter telescope located at the Amundsen–Scott South Pole Station, Antarctica. The telescope is designed
South_Pole_Telescope
Radio observatory in Arecibo, Puerto Rico
Astronomical Society of the Pacific Conference Series. 75: 90–98. Bibcode:1995ASPC...75...90G. "Telescope Description". National Astronomy and Ionosphere
Arecibo_Observatory
Astronomical observatory in Hawaii
observatory with two telescopes at an elevation of 4,145 meters (13,600 ft) near the summit of Mauna Kea in the U.S. state of Hawaii. Both telescopes have 10 m (33 ft)
W._M._Keck_Observatory
Children's musical group and television program
Hi-5, also known as Hi-5 USA, is an American children's television series produced by Kids Like Us for Discovery Kids. It also aired on TLC, on the Ready
Hi-5_(American_TV_series)
Sharpe is a series of historical fiction stories by Bernard Cornwell centred on the character of Richard Sharpe. Cornwell's series (composed of several
List of Sharpe series characters
List_of_Sharpe_series_characters
1999 British TV series or programme
preschool animated children's television series based on the book series of the same name by Lucy Cousins. The series aired from 11 February 1999 to 2 November
Maisy_(TV_series)
Space telescope for gamma-ray astronomy launched in 2008
The Fermi Gamma-ray Space Telescope (FGST, also FGRST), formerly called the Gamma-ray Large Area Space Telescope (GLAST), is a space observatory being
Fermi Gamma-ray Space Telescope
Fermi_Gamma-ray_Space_Telescope
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TELESCOPING SERIES
TELESCOPING SERIES
TELESCOPING SERIES
TELESCOPING SERIES
TELESCOPING SERIES
TELESCOPING SERIES
TELESCOPING SERIES
TELESCOPING SERIES
TELESCOPING SERIES
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