Limit involving Polygamma function: $\lim_{x\to \infty} \sum_{k = 0}^\infty \frac{2x^2}{(x + k)^3}$

Question

I don't remember where I got this identity: $$\lim_{x\to \infty} [(-1)^{k + 1} x^k \psi^{(k)}(x)] = (k - 1)!$$ where $\psi^{(k)}$ is the polygamma function.

I just need to find the limit above for $k = 2$, that is \begin{equation}\lim_{x\to \infty} [-x^2 \psi''(x)]. \tag{*} \end{equation}

I use the series representation of the polygamma function, $$\psi^{(n)}(x) = (-1)^{n + 1} n! \sum_{k = 0}^\infty \frac{1}{(x + k)^{n + 1}},$$ and change $(*)$ to

$$\lim_{x\to\infty}\sum_{k = 0}^\infty \frac{2x^2}{(x + k)^3}.$$

I don't know how to proceed further. Any help would be apreciated. Thanks.

Answer 1

By L'Hôpital's rule, $$\lim_{x\to\infty}\frac{\psi^{(k)}(x)}{(-1)^{k+1}x^{-k}}=\lim_{x\to\infty}\frac{(k-1)!\psi(x)}{\ln x}=(k-1)!.$$