Does the Series ?(?(n+1) – ?n)/n Converge or Diverge? – Sequences and Series

Question

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.

Answer 1

Note that $$\dfrac{\sqrt{n+1} - \sqrt{n}}n = \dfrac1{n(\sqrt{n+1} + \sqrt{n})} < \dfrac1{2n\sqrt{n}}$$

Hence, $$\sum\limits_{n=1}^{\infty}\dfrac1{n(\sqrt{n+1} + \sqrt{n})} < \sum\limits_{n=1}^{\infty}\dfrac1{2n\sqrt{n}} = \dfrac12{\zeta(3/2)} < \infty$$