# Determine if the following series are convergent or divergent?

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. $$\sum_{k=1}^\infty \ln(1+ \sqrt{\frac 2k})$$ $$\sum_{k=1}^\infty\displaystyle \sqrt[k]{e}\sin(\frac{\pi}{k}).$$

• For the second one, can't we use a variation of the harmonic series? – Aranya Lahiri Nov 20 '13 at 19:45

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

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

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