Vault Of Euclid

Sum of a Telescoping Series

Sequences warmup

Investigate the series \( \frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\dots+\frac{1}{n(n+1)}+\dots \)
calculusseriestelescoping-seriespartial-fractionssequence-of-partial-sumsconvergencelimitevaluate-suminfinite-seriesexact-sumrational-termslimits-at-infinity

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What this problem tests

Tests the ability to recognize a telescoping series, apply partial fraction decomposition, and evaluate the limit of a sequence of partial sums.

This is a classic introductory problem in calculus courses when learning about infinite series, demonstrating that some series can be evaluated exactly.

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