How do we find the power series of $f(z) = (1+2z)/(1-2z)$ centered at the origin?

Question

$$f(z) = (1+2z)/(1-2z)$$ Centered at 0.

I was struggling to find its power expansion for f(z)

All I got is $$z<1/2$$ being the only radius of convergence?

$$f(z) = 1/(1-2z) + 2z/(1-2z) = 1/(1-2z) +(-1) 1/(1 - 1/2z)$$

And this is where I get stuck when I try to find its power expansion.

Answer 1

HINT

\begin{align*} \frac{1 + 2z}{1 - 2z} & = \frac{2 - (1 - 2z)}{1 - 2z} =\frac{2}{1 - 2z} - 1 \end{align*}

Now you may consider the geometric series (which converges whenever $|2z| < 1$): \begin{align*} \frac{1}{1 - 2z} = 1 + 2z + (2z)^{2} + (2z)^{3} + \ldots \end{align*}

Can you take it from here?