Homework Help: Series Question

1. Apr 21, 2010

bondi97

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

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

a) Find the Maclaurin Series for f on the doman |z| < 2.

b) Find the Laurent Series for f centered at z0 = 0 on domain 2 < |Z| < inf.

2. Relevant equations

3. The attempt at a solution

I'm having a hard time figuring out how (z + 2)/(z - 2) = 1 + (4/(z-2)) = 1 - (2/(1 - (z/2)).

I tried referring to a geometric series, but I don't think I have the right approach.

Last edited: Apr 21, 2010
2. Apr 21, 2010

lanedance

hmmmm... been a while but how about
$$\frac{z+2}{z-2} = (\frac{1/z}{1/z})\frac{z+2}{z-2} = \frac{1+2/z}{1-2/z} = (1+\frac{2}{z})\frac{1}{1-2/z}$$

3. Apr 21, 2010

lanedance

or for your specific question working back
$$1 + \frac{4}{z-2} = \frac{z-2}{z-2} + \frac{4}{z-2} = \frac{z-2+4}{z-2} = \frac{z+2}{z-2}$$