Q:

Calculus Divergence help with 2 problems

So we got few assignments to practice before our exam, but I'm having a trouble solving 2 of them, would really appreciate any help!
These are the two problems:
The question is do they diverge or not and prove it
$$\sum_{n=1}^{\infty}\frac{1}{n}\bigg(\frac{2}{3-(-1)^n}\bigg)^n$$
$$\sum_{n=1}^{\infty} \bigg(\frac{\sqrt[]{n+1}}{n} - \frac{\sqrt[]{n}}{n+1}\bigg)$$
-I know the first one diverges. (Sorry for my bad english)

A:

The first diverges. Try an even $n$.
The second converges because
$$\frac{\sqrt{n+1}}{n}-\frac{\sqrt{n}}{n+1}=\frac{(n+1)^3-n^3}{n(n+1)(\sqrt{(n+1)^3}+\sqrt{n^3})}=$$
$$=\frac{3n^2+3n+1}{n(n+1)(\sqrt{(n+1)^3}+\sqrt{n^3})}\sim\frac{const}{n^{1.5}}$$ and $1.5>1$.

