## Examine a function - conclutions drawn from taylor series

I'm supposed to examine a function, find the singular part.
$f(z)$ = $\displaystyle \frac{z^2+1}{z(e^{2\pi z} -1)}$

I start with the exponent and do a Taylor expansion.
$\displaystyle e^{2\pi z} -1$ = $\displaystyle 1 + 2\pi z + 2 \pi^2 z^2 + O(z^3) + ... - 1$

All together.
$\displaystyle \frac {z^2 + 1}{z(2\pi z + 2 \pi^2 z^2 + O(z^3) )}$ = $\displaystyle \frac {z^2 + 1}{2\pi z^2}(1 + \pi z + O(z^2) )}$

From this I get
$\displaystyle \frac{1}{2\pi z^2} - \frac{1}{2 z} + \frac{1}{2 \pi} - \frac{z}{2} + ...$

Conclusions drawn from this:
$\displaystyle \frac{1}{2\pi z^2} - \frac{1}{2 z}$ - Singular part

$\displaystyle \frac{1}{2 \pi}$ - ???

$\displaystyle \frac{z}{2} + ...$ - Real part ???

If anyone would enlighten me on what information I can get from a series, and it's parts, like this?