Fining the sum of a series .

Question

I'm solving a probability problem and the expected value of a random variable is this series I need some hint to find the sum of it.

$\displaystyle \sum_{n=1}^{\infty}\frac{n}{2^n}$

differentiate the geometric series. afterwards put $x=1/2$ – tired Mar 05 '17 at 22:00 — tired, Mar 05 '17 at 22:00
Here you have a general technique to evaluate these sums. – Masacroso Mar 05 '17 at 22:01 — Masacroso, Mar 05 '17 at 22:01

score 1 · Answer 1 · answered Mar 05 '17 at 22:09

1

hint

For $x\in(-1,1)$, we have

$$x+2x^2+3x^3+...=x(1+2x+3x^2+...)$$

$$=x\frac{d(x+x^2+x^3+...)}{dx}$$

$$=\frac{x}{1-x}+\frac{x^2}{(1-x)^2}$$

Now, you choose the right $x$.

answered Mar 05 '17 at 22:09

hamam_Abdallah

1 Answers1