Convergence of the series Im trying to resolve the next exercise:
$$\sum_{n=1}^\infty\ e^{an}n^2   \text{  ,  }a\in R $$
I dont know in which ranges I should separe the a value for resolving the limit and finding out the convergence. 

Write it as
$\sum_{n=1}^\infty\ r^n n^2$
where $r = e^a$ satisfies $0 < r$.
If $r \ge 1$ (i.e., $a \ge 0$), the sum clearly diverges.
If $r < 1$ (i.e., $a < 0$), you can get an explicit formula
for $\sum_{n=1}^m\ r^n n^2$
which will show that the sum converges.
Therefore the sum converges for
$a < 0$ and diverges for $a \ge 0$.
The $e^{an}$ seems like a distraction to hide the true nature
of the problem.
