I have the following series:
$$\sum_{n=1}^\infty\frac{2^n-3n}{\sinh(n+2)}x^n$$
And I need to find the values of $x\in\mathbb{R}$ for which the serie converges absolutely, diverges and (if there exists) for which it converges conditionally. $$$$
Divergence: I though about using the n$th$ term test to find values of $x$ for which it diverges, but I can't seem to work around the limit. $$$$
Convergence: I would find values of $x$ for which the sum converges by using the sequence $\frac{2^n}{\sinh(n+2)}x^n$ since it bounds from above the initial one. Then I would use the root test or ratio test but, again, I am not able to work around the $\sinh$.
$$$$ Conditional convergence: I would let $x=-1$ and see if the series convergese using Leibniz Test.
$$$$ Are these ideas correct? Also, can someone explain the evaluation of the limit? Thank you.
[Math] Find for which values of $x$ the series converges
sequences-and-series
Best Answer
As $n$ goes to $+\infty$, we have $2^n-3n\sim 2^n$ and $\sinh(n+2)\sim Ke^n$ where $K>0$.
Hence, if we consider the $n-$th coefficient :
$$a_n=\frac{2^n-3n}{\sinh(n+2)}$$
we have :
$$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\frac2e$$which implies that the radius of convergence of the power series $\sum_{n\ge1}a_nx^n$ is :
$$R=\frac e2$$
The series is absolutely convergent for all $\displaystyle{x\in(-\frac{e}{2},\frac{e}{2})}$, it diverges for $\displaystyle{|x|>\frac e2}$.
It diverges also for $\displaystyle{x\in\{-\frac e2,\frac e2\}}$ since its general term doesn't have zero limit in those cases.