# Infinite series in terms of x

## Homework Statement

Hi,

How do i determine de result in terms of x of this series for x < 1:

(Sum(i=0..+infinity; i*x^i))/(Sum(i=0..+infinity;x^i)

Thanks

## The Attempt at a Solution

I know that (Sum(i=0..+infinity;x^i) will tend do 1/(1-x) but i don't know what the numerator will tend to

Thanks

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tiny-tim
Homework Helper
Hi benf.stokes!

(have a sigma: ∑ and an infinity: ∞ and try using the X2 and X2 tags just above the Reply box )
(Sum(i=0..+infinity; i*x^i))/(Sum(i=0..+infinity;x^i)

I know that (Sum(i=0..+infinity;x^i) will tend do 1/(1-x) but i don't know what the numerator will tend to

Thanks
Hint: integrate.

Thanks, I figured it out but by differentiating (sorry for the delay but I was netless for a few days):

$$\sum_{n=0}^{\infty}x^n=\frac{1}{1-x}$$

$$\sum_{n=0}^{\infty}nx^{n-1}=\frac{1}{(1-x)^2}$$

$$\sum_{n=0}^{\infty}nx^n=\frac{x}{(1-x)^2}$$

How would it be done by integrating? The other way around?

Last edited:
tiny-tim