I have come across the series$$\sum_{n=1}^{\infty} \frac{a^n}{n(n+1)}, \quad|a|<1$$
in an intermediate step while solving a problem.
I wonder if there is a closed form for it?
I could only observe that it can be written as $\sum_{n=1}^{\infty} a^n\left(\frac{1}{n}-\frac{1}{n+1}\right)$, but the usual telescoping trick doesn't work here.
I am unaware of usable techniques. Any help is welcomed.
2026-04-12 17:53:08.1776016388
Closed form of $\sum_{n=1}^{\infty} \frac{a^n}{n(n+1)}$
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