Telescoping series

In mathematics, a telescoping series is a series whose general term ${\displaystyle t_{n}}$ is of the form ${\displaystyle t_{n}=a_{n+1}-a_{n}}$, i.e. the difference of two consecutive terms of a sequence ${\displaystyle (a_{n})}$.

As a consequence the partial sums only consists of two terms of ${\displaystyle (a_{n})}$ after cancellation. The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences.

For example, the series

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n(n+1)}}}$

(the series of reciprocals of pronic numbers) simplifies as

${\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }{\frac {1}{n(n+1)}}&{}=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)\\{}&{}=\lim _{N\to \infty }\sum _{n=1}^{N}\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)\\{}&{}=\lim _{N\to \infty }\left\lbrack {\left(1-{\frac {1}{2}}\right)+\left({\frac {1}{2}}-{\frac {1}{3}}\right)+\cdots +\left({\frac {1}{N}}-{\frac {1}{N+1}}\right)}\right\rbrack \\{}&{}=\lim _{N\to \infty }\left\lbrack {1+\left(-{\frac {1}{2}}+{\frac {1}{2}}\right)+\left(-{\frac {1}{3}}+{\frac {1}{3}}\right)+\cdots +\left(-{\frac {1}{N}}+{\frac {1}{N}}\right)-{\frac {1}{N+1}}}\right\rbrack \\{}&{}=\lim _{N\to \infty }\left\lbrack {1-{\frac {1}{N+1}}}\right\rbrack =1.\end{aligned}}}$

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.