Answer
$\sum_{n=0}^{\infty}(-1)^{n}{4^{n}}(n+1)x^{n+1}$, $R=\frac{1}{4}$
Work Step by Step
$f(x)=\frac{x}{(1+4x)^{2}}=\sum_{n=0}^{\infty}(-1)^{n}{4^{n}}(n+1)x^{n+1}$
$\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\dfrac{(-1)^{n+1}{4^{n+1}}(n+2)x^{n+2}}{(-1)^{n}{4^{n}}(n+1)x^{n+1}}|$
$=|x|\lt \frac{1}{4}$
The given series converges with $R=\frac{1}{4}$