Telescoping Series

Summary

A telescoping series has partial sums in which most intermediate terms cancel, leaving only a few boundary terms. The classic example comes from partial fractions of $1/(n(n+1))$ . Telescoping series are not the same as Taylor series.

Prerequisites

Series Sums by Partial Sums, partial fractions

Definition

A series $\sum a_n$ is telescoping if $a_n=b_n-b_{n+1}$ (or a short fixed-length difference) for some sequence $\{b_n\}$ . Then

and $\sum _{n=1}^{\mathrm{\infty }}{a}_n=\underset{N\to \mathrm{\infty }}{lim}{S}_N=b_1-\underset{N\to \mathrm{\infty }}{lim}{b}_{N+1}$ when the limit exists.

Conditions / Assumptions

• Algebraic cancellation must be valid for the indices used.
• Convergence requires $lim{b}_{N+1}$ to exist (finite).

Worked Example

Partial sums:

Thus $\sum _{n=1}^{\mathrm{\infty }}1/(n(n+1))=1$ .

Common Mistakes

• Calling a Taylor expansion a “telescoping series.”
• Writing the false identity $1/x=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n{x}^n$ as a telescoping series (that sum is the geometric series for $1/(1+x)$ when $|x|<1$ , not $1/x$ ).

Connections

• Geometric Series (another closed-form partial sum)
• Comparison and $p$ -series for related rational terms: P Series, Comparison Tests

References

Telescoping examples appear with partial sums in OpenStax Calculus Volume 2.^[1]