Telescoping series

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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 of the series only consists of two terms of ${\displaystyle (a_{n})}$ after cancellation.^[1][2]

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.^[3]