TITLE: Does the series $\sum_{n=1}^\infty$ ${\sqrt{n+1}-\sqrt{n}}\over n$ converge or diverge?
QUESTION [5 upvotes]: Does this series converge or diverge?
$$\sum_{n=1}^\infty\frac{\sqrt{n+1}-\sqrt{n}}{n}$$
I tried the comparsion test with $\sum_{n=1}^\infty$ $1 \over n$ but this does not help.
I also tried to simplify the series to $\sum_{n=1}^\infty \frac{1}{n (\sqrt{n+1}+\sqrt{n}{}{}{})}$ but this become harder.

REPLY [8 votes]: $\sqrt{1+1/n} < 1+1/(2n)$ (by squaring both sides),
so $\displaystyle\frac{\sqrt{n+1} - \sqrt{n}}{n}
=  \frac{\sqrt{n}(\sqrt{1+\frac{1}{n}} - 1)}{n}
< \frac{\sqrt{n}\frac{1}{2n}}{n}
= \frac1{2n^{3/2}}
$.
Since $\sum \frac1{n^{1+c}}$ converges for any $c > 0$
(by the integral test or many other ways),
the sum converges.