1. ## Laurent series

Consider the identity
$\displaystyle 0 = \frac{1}{z-1} + \frac{1}{1-z} = \frac{1}{z}\frac{1}{1-\frac{1}{z}}+\frac{1}{1-z} = \sum_{n=1}^{infinity} \frac{1}{z^n} + \sum_{n=0}^{infinity}z^n=\sum_{-infinity}^{infinity} z^n$
Does this contradict the uniqueness of the Laurent Series?

due in 15 hours haha
thanks ahead of time if anyone does work on this

2. Originally Posted by jbpellerin
Consider the identity
$\displaystyle 0 = \frac{1}{z-1} + \frac{1}{1-z} = \frac{1}{z}\frac{1}{1-\frac{1}{z}}+\frac{1}{1-z} = \sum_{n=1}^{infinity} \frac{1}{z^n} + \sum_{n=0}^{infinity}z^n=\sum_{-infinity}^{infinity} z^n$
Does this contradict the uniqueness of the Laurent Series?

due in 15 hours haha
thanks ahead of time if anyone does work on this
I don't think the series:

$\displaystyle \sum_{n=-\infty}^{\infty}z^n$ converges anywhere: The singular part converges outside the unit circle; the regular part, inside the unit circle and by casual inspection doesn't seem to converge anywhere on the unit circle.