Evaluate:
$$\sum_{n=1}^{\infty} \frac{H_n}{(n+1)^2}$$
A user stated: "most of the time sum up the residues of $(\gamma+\psi(z))^2\cdot r(z)$. To determine the residues, just expand the digamma function as a Laurent series about the positive integers." where $r(z)$ is the rational function.
Firstly, what is the Laurent series of $\psi(z)$??
And how would I find the residues? Any help is appreciated.
This is a partial answer to your question and I do not know how to get the Laurent series of $\psi(z)$. Note that \begin{eqnarray} \sum_{n=1}^\infty\frac{H_n}{(n+1)^2}&=&\sum_{n=1}^\infty\frac{1}{(n+1)^2}\left(H_n+\frac{1}{n+1}\right)-\sum_{n=1}^\infty\frac{1}{(n+1)^3}\\ &=&\sum_{n=1}^\infty\frac{H_{n+1}}{(n+1)^2}-\sum_{n=1}^\infty\frac{1}{(n+1)^3}\\ &=&\sum_{n=1}^\infty\frac{H_{n}}{n^2}-\sum_{n=1}^\infty\frac{1}{n^3}\\ &=&\sum_{n=1}^\infty\frac{H_{n}}{n^2}-\zeta(3).\\ \end{eqnarray} The latter series can be evaluated using this generating function $$\sum_{n=1}^\infty \frac{H_n\, x^n}{n^2}=\zeta(3)+\frac{\ln^2(1-x)\ln(x)}{2}+\ln(1-x)\operatorname{Li}_2(1-x)+\operatorname{Li}_3(x)-\operatorname{Li}_3(1-x)$$ Taking its limit as $x\to1$, we get \begin{eqnarray} \sum_{n=1}^\infty\frac{H_n}{n^2}=2\zeta(3)\end{eqnarray} and hence $$ \sum_{n=1}^\infty\frac{H_n}{(n+1)^2}=\zeta(3).$$
Start from the singular expansion of the digamma function: $$\psi(z+1)+\gamma = \sum_{n\ge 1} \frac{z}{n(n+z)} = \sum_{n\ge 1} \left(\frac{1}{n}-\frac{1}{z+n}\right).$$
This immediately produces the Laurent expansion at negative integers $-q$ where $q\ge 1$:
$$\psi(z+1)+\gamma = -\frac{1}{z+q} + H_{q-1} + \cdots$$
This is because the sum minus the singular term is $$\frac{1}{q} + \sum_{n=1, n\ne q} \left(\frac{1}{n}-\frac{1}{(z+q)+n-q}\right) \\ = \frac{1}{q} + \sum_{n=1, n\ne q} \left(\frac{1}{n}-\frac{1}{n-q}\frac{1}{(z+q)/(n-q)+1}\right)$$
The constant term here is $$\frac{1}{q} + \sum_{n=1, n\ne q} \left(\frac{1}{n}-\frac{1}{n-q}\right)$$
The outside term and the first term inside the sum iterate over the inverses of all natural $n.$ The second term inside the sum first collects $$\sum_{n=1}^{q-1} \left(-\frac{1}{n-q}\right)$$ which is $H_{q-1}$ and thereafter iterates over the inverses of all natural $n,$ canceling the first contribution, for a final result of $H_{q-1}.$
This immediately implies that $$\mathrm{Res}_{z=-(q-1)} (\psi(z)+\gamma)^2 = -2 H_{q-1}.$$
The sum being evaluated is $$\sum_{n=1}^\infty \frac{H_n}{(n+1)^2} = \sum_{n=2}^\infty \frac{H_{n-1}}{n^2}.$$
Since the residue formula also holds for $q=1$ we can re-write this as $$\sum_{n=1}^\infty \frac{H_{n-1}}{n^2}.$$
Now integrating $$f(z) = \frac{1}{z^2}(\psi(z)+\gamma)^2$$ along a large circle of radius $R$ with $R$ going to infinity and passing between the poles we pick up exactly $-2$ times the value of the sum and the residue at zero.
But at zero we have the expansion $$\psi(z+1)+\gamma = -\sum_{k\ge 1} \zeta(k+1) (-1)^k z^k$$ which yields $$\psi(z)+\gamma = -\frac{1}{z} - \sum_{k\ge 1} \zeta(k+1) (-1)^k z^k.$$
This immediately implies that $$\mathrm{Res}_{z=0} \frac{1}{z^2}(\psi(z)+\gamma)^2 = 2\zeta(3).$$
Because the integral on the circular contour vanishes in the limit (an exercise that is left to the reader that follows from the observation that $2\pi R \times (\log R)^2/R^2$ vanishes in the limit) we get
$$2\zeta(3) - 2 \sum_{n=1}^\infty \frac{H_{n-1}}{n^2} = 0$$ or $$\sum_{n=1}^\infty \frac{H_{n-1}}{n^2} = \zeta(3).$$
The digamma formulae are from the Wikipedia entry.
