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}} , i.e. the difference of two consecutive terms of a sequence ( a n ) {\displaystyle (a_{n})} . As a consequence the partial sums of the series only consists of two terms of ( a n ) {\displaystyle (a_{n})} after cancellation. 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.
Definition
Telescoping sums are finite sums in which pairs of consecutive terms partly cancel each other, leaving only parts of the initial and final terms. Let a n {\displaystyle a_{n}} be the elements of a sequence of numbers. Then
∑ n = 1 N ( a n − a n − 1 ) = a N − a 0 . {\displaystyle \sum _{n=1}^{N}\left(a_{n}-a_{n-1}\right)=a_{N}-a_{0}.}
If a n {\displaystyle a_{n}} converges to a limit L {\displaystyle L} , the telescoping series gives:
∑ n = 1 ∞ ( a n − a n − 1 ) = L − a 0 . {\displaystyle \sum _{n=1}^{\infty }\left(a_{n}-a_{n-1}\right)=L-a_{0}.}
Every series is a telescoping series of its own partial sums.
Examples The product of a finite geometric series with initial term a {\displaystyle a} and common ratio r {\displaystyle r} by the factor ( 1 − r ) {\displaystyle (1-r)} yields a telescoping sum: ( 1 − r ) ∑ k = 0 n a r k = ∑ k = 0 n ( a r k − a r k + 1 ) = a − a r n + 1 {\displaystyle (1-r)\sum _{k=0}^{n}ar^{k}=\sum _{k=0}^{n}\left(ar^{k}-ar^{k+1}\right)=a-ar^{n+1}} When | r | < 1 {\displaystyle |r|<1} , this allows for a direct calculation of its limit as n → ∞ {\displaystyle n\rightarrow \infty } and implies: ∑ k = 0 ∞ a r k = a 1 − r . {\displaystyle \sum _{k=0}^{\infty }ar^{k}={\frac {a}{1-r}}.}
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