Math Help - First four terms of Laurent series

1. First four terms of Laurent series

Find the first 4 terms of the Laurent series centered at z0=0 for 1/ ( e^x - 1)
using division of power series.
Then, give a geometric reason why this expansion is valid on the puctured disk 0 < [z] <2pi

where [z] stands for absolute value of z.

Thank you .

2. Originally Posted by altanl
Find the first 4 terms of the Laurent series centered at z0=0 for 1/ ( e^z - 1)
using division of power series.
Then, give a geometric reason why this expansion is valid on the puctured disk 0 < [z] <2pi

where [z] stands for absolute value of z.

Thank you .
Correction in red.

The first step is to substitute the power series for e^z and simplify. Then divide. Have you done this? Where are you stuck? What have you tried?

By the way, after doing the first step it should be obvious that z = 0 is a simple pole. Therefore the series has the form $\frac{a_{-1}}{z} + a_0 + a_1 z + a_2 z^2 + ....$.

The answer you're aiming for is here: Laurent series of 1&#x2f;&#x28;Exp&#x5b;x&#x5d; - 1&#x29; - Wolfram|Alpha