Find a power series representation for the function
f(x) = x^3 / (x - 2)^2
Ok first we note that
$\displaystyle \frac{d}{dx}\bigg[\frac{1}{x-2}\bigg]=\frac{-1}{(x-2)^2}$
But we also know that
$\displaystyle \frac{1}{x-2}=\frac{-1}{2}\frac{1}{1-\frac{x}{2}}=-\frac{1}{2}\sum_{n=0}^{\infty}\left(\frac{x}{2}\ri ght)^n=-\sum_{n=0}^{\infty}\frac{x^n}{2^{n+1}}$
So we have
$\displaystyle -x^3\cdot\bigg[-\sum_{n=0}^{\infty}\frac{x^n}{2^{n+1}}\bigg]=x^3\cdot\sum_{n=1}^{\infty}\frac{nx^{n-1}}{2^{n+1}}=\sum_{n=0}^{\infty}\frac{nx^{n+2}}{2^ {n+1}}$
$\displaystyle \therefore\frac{x^3}{(x-2)^2}=\sum_{n=0}^{\infty}\frac{nx^{n+2}}{2^{n+1}}$
What do you mean? I multiplied after I took the derivative because if I would have done so before hand I would be saying the following is true
$\displaystyle f(x)\cdot{g'(x)}=\bigg(f(x)\cdot{g(x)}\bigg)'$
Which I hope is not true, or I got an A+ in calculus when I shouldnt have