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This question already has an answer here:

Is there a general rule to find the value of infinite sums like $$\sum\limits_{n=1}^{\infty}\frac{n}{r^n}?$$ I know the formula for a sum of a geometric sequence, but this is a geometric sequence multiplied by an arithmetic sequence. How would one calculate this infinite sum?

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marked as duplicate by Hans Lundmark, mau, user63181, egreg, Moishe Kohan May 30 '14 at 22:29

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  • $\begingroup$ Here is a related question. $\endgroup$ – David Mitra May 30 '14 at 14:29
  • $\begingroup$ This is a polylogarithm. $\endgroup$ – Lucian May 30 '14 at 15:15
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For $\displaystyle|y|<1, \sum_{n=0}^{\infty}y^n=\frac1{1-y}$

Differentiate both sides wrt $y$ and multiply with $y$

Set $\displaystyle y=\frac1x$

See also : Arithmetico-geometric sequence

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  • $\begingroup$ OR $y=\frac 1r$. $\endgroup$ – Thomas Andrews May 30 '14 at 14:50
  • $\begingroup$ @ThomasAndrews, What is $r$ here? $\endgroup$ – lab bhattacharjee May 30 '14 at 15:02
  • $\begingroup$ It's the variable as used in the OPs question. There is no $x$ in the question. $\endgroup$ – Thomas Andrews May 30 '14 at 15:03
  • $\begingroup$ @Downvoter, Please pinpoint the mistake $\endgroup$ – lab bhattacharjee Jun 1 '14 at 14:13
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Hint: Set $x = 1/r$, then $$ \sum_{n=1}^{\infty} nx^n = x \sum_{n=1}^{\infty} nx^{n-1} = x \frac{d}{dx} \sum_{n=0}^{\infty} x^n $$ where the last equality holds when $|x|<1$. Now use your geometric series knowledge.

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