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.
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yoxxxxx
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4Integrate $\sum_1^{\infty} a^{n-1}$ twice and divide by $a$. – Vishu Apr 16 '21 at 15:29
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1Wow it worked thank you very much! – yoxxxxx Apr 16 '21 at 15:32
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Use the formula searching tool approach0.xyz, you will find at once answers such as this one: https://math.stackexchange.com/q/3210006 – Jean Marie Apr 16 '21 at 15:39
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2This becomes popular these days... – metamorphy Apr 16 '21 at 20:55