Find the sum of the series:
- $ \displaystyle \cdots + \frac{1}{z^{3}} + \frac{1}{z^{2}} + \frac{1}{z} + 1 + z + z^{2} + z^{3} \cdots$
This series can be summed in the following way:
$\cdots + \frac{1}{z^{3}} + \frac{1}{z^{2}} + \frac{1}{z} + 1 = \frac{z}{z-1}$ and $ z + z^{2} + z^{3} + \cdots = \frac{z}{1-z}$
So the sum equals $0$.
Is this correct or wrong? Please let me know if there is any error.