What is a telescoping series?

A telescoping series is one where consecutive terms partially cancel, leaving only a few boundary terms — much like how a collapsing telescope shortens. The classic example is sum 1/(n*(n+1)) = sum (1/n - 1/(n+1)), where all middle terms cancel and the infinite sum equals 1.

How is 1/(n*(n+k)) decomposed?

By partial fractions: 1/(n*(n+k)) = (1/k)*(1/n - 1/(n+k)). This splits each term into two fractions that cancel with neighbors when the series is summed.

What does the partial sum S(N) equal?

For starting index n=1: S(N) = (1/k)*(1/1 - 1/(N+k)). As N → ∞, S = 1/k. For starting index n=0 the gap k is used from n=1 onward to avoid division by zero.

Does the series always converge?

Yes. Because the partial sum S(N) = (1/k)*(1/start - 1/(N+k)) and the second term goes to 0 as N grows, the series always converges to (1/k)*(1/start) for any positive integer k.

Can k be any positive integer?

k must be a positive integer (1, 2, 3, …). k=0 would create 1/0 which is undefined, and negative k produces the same series in reverse with a sign change.

Telescoping Series Calculator

Name: Telescoping Series Calculator
Availability: InStock
Author: Nham Vu

Enter a telescoping series of the form 1/(n*(n+k)) to see partial fractions, term-by-term cancellation, and the exact partial and infinite sums.

Series Parameters

Series

∑ 1 / (n × (n + 1))

Gap k (positive integer)

Series: ∑ 1/(n×(n+k))

Starting index n

Terms to show (N)

Between 2 and 20

Term-by-Term Cancellation

n	Term 1/(n(n+k))	Partial fraction	Survives?

Summary

Enter a telescoping series of the form 1/(n*(n+k)) to see partial fractions, term-by-term cancellation, and the exact partial and infinite sums.

How it works

Enter the gap k (the spacing between factors) for the series sum 1/(n*(n+k)).
Choose starting index n = 1 or 0, and the number of terms to display.
The tool applies partial fraction decomposition: 1/(n*(n+k)) = (1/k)*(1/n - 1/(n+k)).
Terms are expanded row by row so you can watch intermediate fractions cancel.
The partial sum S(N) and infinite sum (1/k * 1/start) are displayed.

Use cases

Check homework answers for telescoping series problems.
Visualize why telescoping series converge to a clean closed form.
Verify partial fraction decomposition by expanding the series.
Explore how changing k shifts the cancellation pattern.
Use as a teaching aid to demonstrate series collapsing.
Quickly compute partial sums without manual arithmetic.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu