Trying to find the quotient of infinite sums

[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?

 

[micromass]

Factor x out first, then integrate.

 

[demonelite123]

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