Answer
$f(x)=1+2x+2x^2+2x^3+\ldots$
Interval of convergence: $(-1,1)$
Work Step by Step
$\frac{1}{1-x}=1+x+x^2+x^3+\ldots$ where $|x|<1$....................(1)
Multiply (1) by $x$,
$\frac{x}{1-x}=x+x^2+x^3+x^4+\ldots$ where $|x|<1$.................(2)
Adding the two series above,
$\frac{1+x}{1-x}=1+2x+2x^2+2x^3+\ldots$ where $|x|<1$.
Thus, we get $f(x)=1+2x+2x^2+2x^3+\ldots$ with the interval of convergence $(-1,1)$
