Finding the Taylor Series of (1+z)/(1-z) for |z|<1

[adjklx]

Homework Statement

Find the taylor series of $$\frac{1+z}{1-z}$$ where $$z$$ is a complex number and $$|z| < 1$$

Homework Equations

$$\sum^{\infty}_{0} z^n = \frac{1}{1-z}$$ if $$|z| < 1$$

The Attempt at a Solution

$$\sum^{\infty}_{0} z^n = \frac{1}{1-z}$$

$$\frac{1+z}{1-z} = \sum^{\infty}_{0} z^n * (1+z) = \sum^{\infty}_{0} z^n + z^{n+1}$$

I was wondering if this is as far as you can go, or if there is a more simple closed form expression for this

 

[gabbagabbahey]

Divide the sum into two parts and notice that

$$\sum_{n=0}^{\infty} z^{n+1}=\left( \sum_{n=0}^{\infty} z^{n} \right) - 1$$