Is the series $\frac{1}{(n+1)^p}-\frac{1}{(n-1)^p}$ where 0<p<1 convergent or divergent?

Question

Sorry 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……

Answer 1

Notice that if

$$a_n = \frac{1}{n^p} $$

The series you're asking about it equivalent to

$$ \sum_{n=2}^\infty a_{n+1} - a_{n-1} $$

Hint: It's telescoping.

Answer 2

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.