Evaluating $\sum\limits_{n=1}^{\infty}\frac{n}{r^n}$

Question

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?

Here is a related question. – David Mitra May 30 '14 at 14:29 — David Mitra, May 30 '14 at 14:29

score 1 · Answer 1 · answered May 30 '14 at 14:29

1

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

answered May 30 '14 at 14:29

lab bhattacharjee

274,582

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
@Downvoter, Please pinpoint the mistake – lab bhattacharjee Jun 01 '14 at 14:13

score 1 · Answer 2 · answered May 30 '14 at 14:31

1

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.

answered May 30 '14 at 14:31

Tom

9,978

Evaluating $\sum\limits_{n=1}^{\infty}\frac{n}{r^n}$

2 Answers2