Telescoping Series

The central idea

A telescoping series has partial sums in which most terms cancel after expansion, often following a decomposition $a_n=b_n-b_{n+1}$. Convergence and sum are determined by writing the finite partial sum $S_N$ first and then taking $N\to\infty$.

Definitions, hypotheses, and notation

When the shift exceeds one, several initial and final terms survive. For $b_n-b_{n+k}$, the first $k$ positive terms remain and are balanced by $k$ tail terms. Writing at least enough expanded terms to show the shift prevents accidental over-cancellation.

A telescoping series may still diverge if the surviving boundary expression lacks a finite limit. Conversely, not all convergent series telescope. The technique is an exact summation structure, so use it when cancellation is visible rather than as a generic convergence test.

Conceptual meaning

Cancellation is an exact finite phenomenon before it is a limiting one. Interior terms appear once positively and once negatively, leaving a few boundary terms. The surviving tail boundary controls the infinite sum.

A dependable method and decision rule

1. Use algebra or partial fractions to expose a difference pattern.
2. Write several terms with indices visible.
3. Form the finite partial sum through $N$.
4. Cancel only terms actually present and identify survivors.
5. Take the limit of the resulting $S_N$ formula.

Fully worked example

Common mistakes and why they fail