Professor really hasn't explained any of this and gave us a practice exam on it.

$\displaystyle f(x)=\frac{x}{4x+1}$

Would this simply be $\displaystyle \sum_{k=0}^{\infty}x(-1)^{k}(4x)^{k}=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{4}(4x)^{k+1}$

I reasoned this since:

$\displaystyle \frac{1}{1+4x}=\sum_{k=0}^{\infty}(-1)^{k}(4x)^{k}$