# Evaluating $\sum\limits_{n=1}^{\infty}\frac{n}{r^n}$ [duplicate]

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?

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

• Here is a related question. – David Mitra May 30 '14 at 14:29
• This is a polylogarithm. – Lucian May 30 '14 at 15:15

For $\displaystyle|y|<1, \sum_{n=0}^{\infty}y^n=\frac1{1-y}$

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

Set $\displaystyle y=\frac1x$

• OR $y=\frac 1r$. – Thomas Andrews May 30 '14 at 14:50
• @ThomasAndrews, What is $r$ here? – lab bhattacharjee May 30 '14 at 15:02
• It's the variable as used in the OPs question. There is no $x$ in the question. – Thomas Andrews May 30 '14 at 15:03
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.