# Finding a taylor series

1. Oct 11, 2008

### adjklx

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

3. 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

2. Oct 11, 2008

### gabbagabbahey

Divide the sum into two parts and notice that

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

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