I was calculating the Laurent series of $\frac{z}{z^2+1}$ centered in $0$ and that converges for $|z|>3$ when in the middle of my calculations I wondered why the follwing is false:
$$\begin{align} &\sum_{k=0}^\infty (-1)^kz^{-2k-1}=&\left[\text{Let }m=-2k-1<=>k=-\frac{m+1}{2}\right]\\ &=\sum_{m=-\infty}^{-1}(-1)^{-\frac{m+1}{2}}z^m=\\ &=\sum_{-\infty}^{-1}i^{-(m+1)}z^m=\\ &=\sum_{-\infty}^{-1}\frac{z^m}{i^{m+1}} \end{align}$$
When I start doing calculations with this Laurent series, those calculations turned out to be incorrect. Why isn't this a correct manipulation?
In the first sum an important property is that it only has odd exponentials. As an example the coefficient for $z^{-2}$ is zero and so forth. However this property is not preserved when I do the manipulation $[\text{Let }=−2−1<=>=−\frac{+1}{2}]$ since I get a square root. As an example the coefficient for $z^{-2}$ therefore becomes $\frac{1}{i}\neq0$. A better manipulation would be $m=-k$.