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| &lt; 1

Homework Equations

<br /> \sum^{\infty}_{0} z^n = \frac{1}{1-z} if |z| &lt; 1<br />

The Attempt at a Solution

<br /> \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}<br />

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