$f(x) = \sum_{k=1}^{\infty} (\frac{1}{x+k} - \frac{1}{x-k})$

Question

I am trying to find a closed form for this sum $$f(x) = \sum_{k=1}^{\infty} (\frac{1}{x+k} - \frac{1}{x-k})$$

Based on a recent answer that Felix Marin posted to my question, this is what I got:

$$\lim_{n \to \infty} (H_{x+n} - H_{-x+n}) - (H_{x-1} - H_{-x-1})$$

I am new to this and don't know how to proceed. Any help? I have a feeling that FM's answer already has a solution to this, but I am just not able to get there.

But can it be expressed something like an impulse train (dirac-delta comb)? — Srini, Jul 04 '17 at 19:07

score 0 · Accepted Answer · answered Jul 04 '17 at 19:10

If Harmonic number function is $H_x$ then the sum is $$-H_{-x}-H_x+2 \gamma$$ Shifting the index by one we have $$\sum _{k=0}^{\infty } \left(\frac{1}{k+x+1}+\frac{1}{k-x+1}\right)$$ Using digamma $\psi$ function we have $$-\psi(x+1)-\psi(-x+1)=-H_{-x}-H_x+2 \gamma$$

see here

$f(x) = \sum_{k=1}^{\infty} (\frac{1}{x+k} - \frac{1}{x-k})$

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