Determine if the following series are convergent or divergent?

Question

How to determine if the following series are convergent or divergent? I'm supposed to use here the limit comparison test, but I don't know how to choose the second series. $$\sum_{k=1}^\infty \ln(1+ \sqrt{\frac 2k})$$ $$\sum_{k=1}^\infty\displaystyle \sqrt[k]{e}\sin(\frac{\pi}{k}).$$

Answer 1

A related problem. Since

$$ \lim_{k\to \infty }\frac{\ln(1+\sqrt{2/k})}{\sqrt{2/k}}=1, $$

then the series diverges by the fact:

Suppose $\sum_{n} a_n$ and $\sum_n b_n $ are series with positive terms, then

if $\lim_{n\to \infty} \frac{a_n}{b_n}=c>0$, then either both series converge or diverge.

Note: We used the Taylor series

$$ \ln(1+t)=t+O(t^2)\implies \ln(1+t)\sim t. $$

$$e^t\sin(\pi t)= \pi t+O(t^2)\implies e^t\sin(\pi t) \sim \pi t.$$