I've been stuck with calculating the sum of series of the following problem. Can you help me?
$$\sum_{n=1}^\infty \frac{x+a}{n(x+a) + n^2}$$
for real numbers $a\geq 0$ and $x\geq 1$.
2026-04-01 11:29:49.1775042989
Calculate sum of series $\sum_{n=1}^\infty \frac{x+a}{n(x+a) + n^2}$.
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After @URL's comment, using $$S_p=\sum_{n=1}^p\frac{x+a}{n(x+a)+n^2}=\sum_{n=1}^p\frac1n-\sum_{n=1}^p\frac1{n+(x+a)}$$ and using generalized harmonic numbers, we have $$S_p=H_{a+x}+H_p-H_{a+x+p}$$ Now, using the asymptotics $$H_q=\gamma +\log \left({q}\right)+\frac{1}{2 q}-\frac{1}{12 q^2}+O\left(\frac{1}{q^3}\right)$$ and using it for the second an third term, we end with $$S_p=H_{a+x}-\frac{a+x}{p}+O\left(\frac{1}{p^2}\right)$$