Exercise 198Calculus › Single-variable calculus › SequencesHigh schoolAdded 2 Nov 2026

A telescoping sum

#telescoping#series#partial sums

Statement

Let $u_n=\dfrac1{n+1}-\dfrac1{n+2}$ for $n\ge0$ . Compute $S_n=\sum_{k=0}^nu_k$ and deduce the limit of $S_n$ .

Hint

Write out the first few terms and watch what cancels.

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Solution

$S_n=\left(1-\frac12\right)+\left(\frac12-\frac13\right)+\dots+\left(\frac1{n+1}-\frac1{n+2}\right)=1-\frac1{n+2},$
because every intermediate term appears once with each sign. As $n\to\infty$ , $S_n\to1$ .

A telescoping sum

Statement

Solution

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