telescoping series

A telescoping series is one whose terms are deliberately differences, so that when you add them up almost everything cancels — like a collapsible spyglass sliding shut, leaving only the ends. The middle terms annihilate in adjacent pairs, and the partial sum reduces to a tidy expression in just the first and last surviving pieces.

Formally, suppose a_n = b_n - b_{n+1} for some sequence b_n. Then the partial sum telescopes: s_N = (b_1 - b_2) + (b_2 - b_3) + ... + (b_N - b_{N+1}) = b_1 - b_{N+1}. The series sum a_n therefore converges if and only if b_{N+1} has a limit as N -> infinity, and when it does, the sum equals b_1 - lim b_N. This turns a convergence question about a series directly into a limit question about the auxiliary sequence b_n.

Telescoping is one of the rare situations where an exact sum is obtainable rather than just a convergence verdict, so it is prized in examples and in deriving closed forms. The art lies in recognizing or engineering the difference structure, often via partial fractions; not every series telescopes, but many textbook sums secretly do.