Trying to find the quotient of infinite sums

1. Feb 7, 2012

demonelite123

i am trying to re-express the following in terms of a rational function: $\frac{(0+x+2x^2+3x^3+...)}{1+x+x^2+x^3+...}$. i know that this is supposed to be $\frac{1}{x-1}$ but I can't figure out how to do it.

I know the denominator is just $\frac{1}{1-x}$. so in order for this work out, the infinite sum which makes up the numerator should be $\frac{1}{(1-x)(x-1)}$. so my problem is figuring out how to express $0+x+2x^2+3x^3+...$ as a function of x. I have tried integrating/differentiating the series which didn't work and i haven't been able to figure out another way to do this.

can someone help me figure this out?

2. Feb 7, 2012

micromass

Staff Emeritus
Factor x out first, then integrate.

3. Feb 8, 2012

demonelite123

oh wow I am really getting rusty on my calculus. thanks for your reply!