Determine if the following series are convergent or divergent?
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
$$ \ln(1+t)=t+O(t^2)\implies \ln(1+t)\sim t. $$
$$e^t\sin(\pi t)= \pi t+O(t^2)\implies e^t\sin(\pi t) \sim \pi t.$$