I have to study for which values $\alpha\in\mathbb R$, the following sum converges $\forall x\in(2,3)$:
$$\overset{\infty}{\underset{n=1}{\sum}}\dfrac{\log(1+e^{\alpha n})}{1+n^{\alpha}}(x-e^{\alpha})^n.$$
I made a substitution $u=x-e^{\alpha }$ and I've studied the succession $a_n:=\dfrac{\log(1+e^{\alpha n})}{1+n^{\alpha}}$ in order to calculate the radius of convergence of the series $\overset{\infty}{\underset{n=1}{\sum}a_nu^n}$ .
Omitting the symbol of limit, I found $\bigg(\dfrac{\log(1+e^{\alpha n})}{1+n^{\alpha}}\bigg)^{\frac{1}{n}}=e^{\log\Big({\Big(\frac{\log(1+e^{\alpha n})}{1+n^{\alpha}}\Big)^{\frac{1}{n}}}\Big)}=e^{\frac{\log(\log(1+e^{\alpha n}))-\log(1+n^{\alpha})}{n}}=e^{\frac{\log(\log(1+e^{\alpha n}))}{n}-\frac{\log(1+n^{\alpha})}{n}}$.
Now I think that, if $\alpha$ is positive, the order of infinity of $n$ is greater than the order of infinity of the $\log$ term. I tried to study the case $\alpha <0$ and I wrote the Taylor's polynomial for $\log(1+e^{\alpha n})$ and $\log(1+n^{\alpha})$, since $n^{\alpha},e^{\alpha n}\underset{n\to\infty}{\longrightarrow}0$.
The problem is that I don't manage to find a useful condition for the parameter.
Furthermore, studying $2<u<3$, I got the condition $\alpha\in[\log 2,\log3]$ but I can't find other conditions (I know there are other conditions because I know the risult)...
Thank you in advance.