General formula for $\sum_{n=1}^{\infty} n^k r^n$ for arbitrary $k$

Asked Oct 20 '18 at 08:11

Active Oct 20 '18 at 08:24

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Is there a general formula for the above summation (in terms of $r$ and $k$)? I've read a very helpful answer on the general formula for $\sum_{n=1}^{\infty} n r^n$, and according to this answer, there was a general formula for arbitrary $k$; however, I couldn't find any references in there as to what it was.

Thanks in advance, all your help is greatly appreciated!

asked Oct 20 '18 at 08:11

QingQuan Xia

1

Per definitionem this is the polylog function $\text{Li}_{-k}(r)$. See e.g. http://mathworld.wolfram.com/Polylogarithm.html – Dr. Wolfgang Hintze Oct 20 '18 at 08:41
Oh, I didn't notice that. Thanks! – QingQuan Xia Oct 21 '18 at 08:18
Are $k$ and $r$ both integers? – Mostafa Ayaz Nov 30 '18 at 18:59

General formula for $\sum_{n=1}^{\infty} n^k r^n$ for arbitrary $k$

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