convergence of $\sum_{n=1}^{\infty}(n^2+1)\log^\alpha \left ( \frac{ \sinh \frac{1}{n}}{\sin\frac{1}{n}} \right )$ $ \sum_{n=1}^{\infty}(n^2+1)\log^\alpha \left ( \frac{\sinh\frac{1}{n}}{\sin\frac{1}{n}} \right ) $
Find the value of $\alpha$ for which the series converges
please, I have no idea to approach the solution
thanks.

Hint. By using Taylor series expansions, as $n \to \infty$, we have
$$
\frac{\sinh \frac1n}{\sin \frac1n} = 1+\frac{1}{3 n^2}+O\left(\frac{1}{n^4}\right)
$$ and
$$
(n^2+1)\log^\alpha \left(\frac{\sinh\frac{1}{n}}{\sin\frac{1}{n}}\right)=\frac{1}{3^a n^{2\alpha-2}}+O\left(\frac{1}{n^{2\alpha}}\right)
$$ the given series is then convergent iff we have $2\alpha-2>1$.
