What is the Result of an Infinite Series in Terms of x for x < 1?

[benf.stokes]

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

 

[tiny-tim]

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.

 

[benf.stokes]

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

<br /> \sum_{n=0}^{\infty}x^n=\frac{1}{1-x}<br />

<br /> \sum_{n=0}^{\infty}nx^{n-1}=\frac{1}{(1-x)^2}<br />

<br /> \sum_{n=0}^{\infty}nx^n=\frac{x}{(1-x)^2}<br />

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

 

[tiny-tim]

Sum the integral, and then differentiate that sum.