Telescoping series

Summary

A telescoping series is a series whose terms have the form $a_n = b_n - b_{n+1}$ for a sequence $(b_n)$. Its partial sums collapse: $\sum_{n=0}^{N} (b_n - b_{n+1}) = b_0 - b_{N+1}$. Thus the series converges if and only if $(b_n)$ converges, and then its sum is $b_0 - \lim b_n$.

Remarks

For example, $\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$, so $\sum_{n=1}^{\infty} \frac{1}{n(n+1)} = 1$. The finite form $\sum_{k=0}^{n-1} (f(k+1) - f(k)) = f(n) - f(0)$ is a discrete analogue of the fundamental theorem of calculus. Mathlib states it as Finset.sum_range_sub.

Explanations

L1 human

In a telescoping sum, each term cancels part of the next term, like the segments of a telescope that slide into each other. Only the first piece and the last piece survive.