Convergence of series: $\log\left(1 + \frac1{n\sqrt{n}}\right )$

Question

I need to find if this series converges or diverges:

$$\sum_{n=1}^\infty \log\left(1+{1\over(n\sqrt{n})}\right)$$

I'm taking calculus III, so I'm allowed to use these test: nth root test, ratio test, nth term test, limit comparison test, direct comparison test, alternating series test.

Answer 1

Alright so I use this $0<ln(1+{1\over(n\sqrt{n}})< 1/(n\sqrt n)$ and then I compare using Direct Comparison Test. Which gives us that the series converges since the $\sum_{n=1}^\infty1/(n\sqrt n)$ is a geometric p-series with p>1.

Answer 2

HINT:

$$0<\log(1+x)\le x$$

for $x>0$. Now use the comparison test.

Answer 3

Such series is obviously less than $$ \sum_{n\geq 1}\frac{1}{n^{3/2}}=\zeta\left(\frac{3}{2}\right) $$ and obviously greater than $$ \log(2)+\log\prod_{p\in\mathcal{P}}\left(1+\frac{1}{p^{3/2}}\right)=\log\left(\frac{2\,\zeta(3/2)}{\zeta(3)}\right). $$