study of the convergence of the numerical series

Question

Can anyone tell me if I answered correctly the convergence of this series $$\sum_{i=1}^\infty \frac{\sqrt{i+1} - \sqrt{i}}{i}$$ $$= \sum_{i=1}^\infty \frac{1}{i(\sqrt{i+1} + \sqrt{i})} = \sum_{i=1}^\infty \frac{1}{i ((i+1)^{1/2} + i^{1/2})} = \sum_{i=1}^\infty \frac{1}{i (i+1)^{1/2} + i^{3/2}}$$
$$i (i+1)^{1/2} + i^{3/2} \sim i^{3/2} \Rightarrow \sum_{i=1}^\infty \frac{1}{i^{3/2}} = \sum_{i=1}^\infty \biggl(\frac{1}{i}\biggr)^{3/2}$$
$\bigl(\frac{1}{i}\bigr)^{3/2}$ is a general harmonic series with $\alpha$ > 1 so the series converges. I don't know if I applied correctly the asymptotic relationship (I don't know if it's called like that in English) in the denominator.

Answer 1

We have that

$$\frac{1}{i(\sqrt{i+1} + \sqrt{i})}\sim \frac{1}{2i^{3/2}}$$

then refer to limit comparison test with $\sum \frac{1}{i^{3/2}}$.