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