Show that $g(z)$ is a function analytic on region $D$, if $f(z)$ is analytic on $D$ and $|f(z)| < 1$, where $g(z) = f(z) + 2(f(z))^2 + 3(f(z))^3+...$

Question

Show that $g(z)$ is a function analytic on region $D$, if $f(z)$ is analytic on $D$ and $|f(z)| < 1$, where $g(z) = f(z) + 2(f(z))^2 + 3(f(z))^3+\dots$

I proved $g(z)$ exists, i.e. infinite sum is convergent. But I can't prove $g(z)$ analytic.

Answer 1

Hint: for $|w|<1$ we have $\sum_{n=1}^{\infty}nw^n=\frac{-w}{(1-w)^2}$

Edit: sorry, a little mistake. The correct result is $\frac{w}{(1-w)^2}$