What is the convergence of the series of $u_n=\frac{1}{(n-1)^\alpha}+\frac{1}{(n+1)^\alpha}-\frac{2}{n^\alpha}$ when $\alpha$ isn't $1$ or $0$?

Question

let be $u_n=\frac{1}{(n-1)^\alpha}+\frac{1}{(n+1)^\alpha}-\frac{2}{n^\alpha}$.

Examine the convergence of these series.

I distinguished three cases the first $\alpha=0$ and $\alpha=-1$ are obvious (simple convergence towards 0 and divergence). but what about the third one? I'm thinking about Taylor expansion but I don't know which one...

Answer 1

\begin{align} &\sum_{n=2}^{\infty}\left(\frac{1}{(n-1)^\alpha}+\frac{1}{(n+1)^\alpha}-\frac{2}{n^\alpha}\right)\\ &=\sum_{n=2}^{\infty}\left(\frac{1}{(n-1)^\alpha}-\frac{1}{n^\alpha}\right)-\sum_{n=2}^{\infty}\left(\frac{1}{n^\alpha}-\frac{1}{(n+1)^\alpha}\right)\\ &=1-\lim_{n\to\infty}\frac{1}{n^\alpha}-\frac12+\lim_{n\to\infty}\frac{1}{(n+1)^\alpha} \end{align} converges if $\alpha\ge0$.