Find all x for which this series converges: from n=0 to infinity, the summation of ((-1)^n)(x)^n/(4)^n?
$\displaystyle \sum_{n=0}^{\infty} (-1)^n \left( \frac{x}{4} \right)^n $ will converge (by the alternating series test) if $\displaystyle \left| \frac{x}{4} \right|^n \rightarrow 0$ as $\displaystyle n \rightarrow \infty$ .....