Exercise 391Calculus › Single-variable calculus › Series › Convergence testsL1Added 2 Nov 2026

Telescoping with square roots

#telescoping#square roots#divergence#p-series

Statement

Study the series $\displaystyle\sum_{n\ge1}\left(\sqrt{n+1}-\sqrt n\right)$ .

Hint

Compute the partial sum directly, and compare with the rationalised form $\frac1{\sqrt{n+1}+\sqrt n}$ .

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Solution

Direct. The partial sums telescope: $S_N=\sqrt{N+1}-1\to+\infty$ . The series diverges.

By equivalents. $\sqrt{n+1}-\sqrt n=\dfrac1{\sqrt{n+1}+\sqrt n}\sim\dfrac1{2\sqrt n}$ , and $\sum\frac1{\sqrt n}$ diverges ( $p=\frac12\le1$ ). Same conclusion.

Remark. The first method is more informative: the partial sums grow like $\sqrt N$ , exactly like $\int_1^N\frac{dt}{2\sqrt t}$ .

Telescoping with square roots

Statement

Solution

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