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Evaluating a Telescoping Series with Partial Fractions

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Study the series \( \sum_{n=1}^{\infty} \frac{1}{n(n+4)} = \frac{1}{1\cdot 5}+\frac{1}{2\cdot 6}+\frac{1}{3\cdot 7}+\dots \)
infinite-seriestelescoping-seriespartial-fractionsconvergenceexact-sumsequence-of-partial-sumslimit-of-sequencecalculus-iialgebraic-manipulation

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What this problem tests

Tests the ability to recognize a telescoping series, perform partial fraction decomposition, and correctly identify which terms survive the cancellation.

This is a standard problem in Calculus II when introducing infinite series and techniques for finding exact sums.

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