Finding the values of $x$ where the series converges absolutely and conditionally

Question

So I have a question about the convergence of a series here:

It says:

Find the radius and interval of convergence, then identify the values of $x$ for which the series converges absolutely and conditionally

$\displaystyle \sum_{n=1}^{\infty} (\text{csch}(n))x^n$

I already found the radius of convergence. It's $e$. The interval of convergence is $(-e,e)$. I checked the endpoints and they both diverge.

So I know for all values of $x$ on the interval $(-e,e)$, the series converges absolutely. What about the conditional convergence though? How do I show that part? I'm a little confused on that. I don't even think that would ever happen.

Answer 1

The series is not convergent at $x=\pm e$ [ Note that $\frac {e^{n}} {e^{n}-e^{-n}}=\frac 1 {1-e^{-2n}}$ which does not tend to $0$ as $ n \to \infty$. A series $\sum a_n$ cannot converge (even conditionally) unless $a_n \to 0$.