Q:

for what $a$ values does $\sum_{n=1}^{\infty}{(\frac{an}{n +1})^n} , a\in \mathbb{R}$ converge?

for what values of a does this series converge\diverge, absolutely converge diverge
$\sum_{n=1}^{\infty}{(\frac{an}{n
+1})^n} , a\in \mathbb{R}$

at first i wanted to do the root test, but the problem is that $a$ could be negative,
if i take it out of the fraction to get $(\frac{a}{n+1})^n$ it doesent meet the critiria of either Dirichlet, or Abel tests

A:

If $a\ne0$ then 
$$\left(\frac{an}{n+1}\right)^n=a^n \cdot \left(\frac{n}{n+1}\right)^n = a^n \cdot \left(
\frac{n+1}{n} \right)^{-n}
=a^n\left(1+\frac{1}{n}\right)^{-n}\sim e^{-1}a^n$$
so the given series is convergent if and only if $|a|<1$. The result is clear for $a=0$.

