Is the series $\frac{1}{(n+1)^p}-\frac{1}{(n-1)^p}$ where 0Sorry for my bad English.
I really suspect it is convergent. But I can't prove it.
Since ${x^p}$ is not derivable at x=0, I can't using taylor expansion to find the order of infinitesimal, thus nth-term test cannot be used. I tried other test but they seem to  lead to a very complex expression.
Is the series convergent or divergen, and how to prove it ?
P.S. sorry for asking such a stupid question……

Since the OP has already the answer, I'll develope the whole computation for future visitors.
The $k$th partial sum is (beginning at $n=2$):
$$3^{-p}-1+4^{-p}-2^{-p}+\cdots+(k+1)^{-p}-(k-1)^{-p}=-1-2^{-p}+k^{-p}+(k+1)^{-p}$$
The limit of this expression when $k\to\infty$ is
$$-1-\frac1{2^p}$$
so the series converges and this is its limit.
