Originally Posted by

**symstar** My real question does not have to do with the problem itself so much as it has to do with a concept within problems like these... or, perhaps, series in general.

$\displaystyle f(x)=\frac{x^3}{(x-2)^2}$

Breaking it down into pieces, starting first with the following...

$\displaystyle \frac{1}{x-2}= \sum_{n=0}^{\infty} \frac{1}{2^{n+1}}x^n$

Now to the next step (where my problems begin)...

$\displaystyle \frac{1}{(x-2)^2}$

$\displaystyle =\frac{d}{dx}\Big[\frac{1}{x-2}\Big]$

$\displaystyle =\frac{d}{dx}\Big[$$\displaystyle \sum_{n=0}^{\infty} \frac{1}{2^{n+1}}x^n\Big]$

And now my problem...

$\displaystyle =\sum_{n=1}^{\infty}$$\displaystyle \frac{1}{2^{n+1}}nx^{n-1}$

My problem is in not knowing how we went from starting the sum at n=0 to suddenly starting it at n=1.

Why does this happen and when should it happen?