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

I know the denominator is just [itex] \frac{1}{1-x}[/itex]. so in order for this work out, the infinite sum which makes up the numerator should be [itex]\frac{1}{(1-x)(x-1)}[/itex]. so my problem is figuring out how to express [itex]0+x+2x^2+3x^3+...[/itex] 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?

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# Trying to find the quotient of infinite sums

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