Convergence of series with logarithms and square roots

Question

I. $\sum_{n=1}^\infty \ln \left(1 + \frac{1}{n^2}\right)$

II. $\sum_{n=1}^\infty \left( \sqrt{n^3} - \sqrt{n^3 - 1} \right)$

I honestly have no idea as to how to approach both these problems. By "observation" I feel like both would diverge, but they both actually converge. Any help would be appreciated.

Also, I would like to know how to approach series with logarithms and square roots in general. I can't think of what method could be used for these types of series.

Answer 1

HINT

For the first note that

$$\ln (1 + \frac{1}{n^2})\sim \frac{1}{n^2}$$

and for the second

$$\sqrt{n^3} - \sqrt{n^3 - 1}=\frac{1}{\sqrt{n^3} + \sqrt{n^3 - 1}}\sim \frac1{2n^{\frac32}}$$

then refer for to limit comparison test.