Editing Telescoping series (section)

{{Short description|Series whose partial sums eventually only have a fixed number of terms after cancellation}}
{{Ref improve|date=March 2021}}
In [[mathematics]], a '''telescoping series''' is a [[series (mathematics)|series]] whose general term <math>t_n</math> is of the form <math>t_n=a_{n+1}-a_n</math>, i.e. the difference of two consecutive terms of a [[sequence]] <math>(a_n)</math>. As a consequence the partial sums of the series only consists of two terms of <math>(a_n)</math> after cancellation.<ref name=":0">{{cite book |last1=Apostol |first1=Tom |title=Calculus, Volume 1 |date=1967 |publisher=John Wiley & Sons |edition=Second |pages=386–387 |orig-date=1961}}</ref><ref>Brian S. Thomson and Andrew M. Bruckner, ''Elementary Real Analysis, Second Edition'', CreateSpace, 2008, page 85</ref> 

The cancellation technique, with part of each term cancelling with part of the next term, is known as the '''method of differences'''.

An early statement of the formula for the sum or partial sums of a telescoping series can be found in a 1644 work by [[Evangelista Torricelli]], ''De dimensione parabolae''.<ref>{{cite book
 | last = Weil | first = André | author-link = André Weil
 | editor1-last = Aubert | editor1-first = Karl Egil | editor1-link = Karl Egil Aubert
 | editor2-last = Bombieri | editor2-first = Enrico | editor2-link = Enrico Bombieri
 | editor3-last = Goldfeld | editor3-first = Dorian | editor3-link = Dorian M. Goldfeld
 | contribution = Prehistory of the zeta-function
 | doi = 10.1016/B978-0-12-067570-8.50009-3
 | location = Boston, Massachusetts
 | mr = 993308
 | pages = 1–9
 | publisher = Academic Press
 | title = Number Theory, Trace Formulas and Discrete Groups: Symposium in Honor of Atle Selberg, Oslo, Norway, July 14–21, 1987
 | year = 1989}}</ref>