Is the series converges or diverges? $\frac{n}{n+1}^{(n^{\frac{3}{2}})}$

Question

$\sum_{n\geq 1}^{}\frac{n}{n+1}^{n^{\frac{3}{2}}}$

I need some help please, I have no idea where to approach the solution

Answer 1

By observing that, as $n \to \infty$, $$ \ln \left(1+\frac1n \right) \sim \frac1n $$ one may write, as $n \to \infty$, $$ \left ( \frac{n}{n+1} \right )^{\large n^{3/2}}=e^{-\large n^{3/2}\ln \left(1+\frac1n \right)} \sim e^{-\sqrt{n}} $$ which is a general term of a convergent series, as one may sees by using $$ \lim_{n \to \infty}n^2 \cdot e^{-\sqrt{n}}=0. $$