Editing Telescoping series (section)

== Definition ==
[[File:Telescoping Series.png|right|thumb|350px|A telescoping series of powers. Note in the [[summation sign]], <math display="inline">\sum</math>, the index ''n'' goes from 1 to ''m''. There is no relationship between ''n'' and ''m'' beyond the fact that both are [[natural numbers]].]]

Telescoping [[sum (mathematics)|sums]] are finite sums in which pairs of consecutive terms partly cancel each other, leaving only parts of the initial and final terms.<ref name=":0" /><ref>{{cite web |last1=Weisstein |first1=Eric W. |title=Telescoping Sum |url=https://mathworld.wolfram.com/TelescopingSum.html |website=MathWorld |publisher=Wolfram |language=en}}</ref> Let <math>a_n</math> be the elements of a sequence of numbers. Then
<math display="block"> \sum_{n=1}^N \left(a_n - a_{n-1}\right) =  a_N - a_0.</math>
If <math> a_n </math> converges to a limit <math>L</math>, the telescoping [[Series (mathematics)|series]] gives:
<math display="block"> \sum_{n=1}^\infty \left(a_n - a_{n-1}\right) = L-a_0. </math>

Every series is a telescoping series of its own partial sums.<ref name=":3">{{Cite book |last1=Ablowitz |first1=Mark J. |title=Complex Variables: Introduction and Applications |last2=Fokas |first2=Athanassios S. |publisher=Cambridge University Press |year=2003 |isbn=978-0-521-53429-1 |edition=2nd |pages=110}}</ref>