I think this was already asked but I couldn't find a question for a function with this kind of problem on the point where the Taylor series are developed around. I see on my text on complex variables (and on computation engines) that the Maclaurin series of the function $f(z) = e^z/z^2$ is written as
$$f(z) = \frac{1}{z^2} + \frac{1}{z} + \frac{1}{2} + \frac{z}{3!} + \frac{z^2}{4!} + \cdots$$
But since the first term of the series is $f(0)$ I don't know how they get these series for this function. I also see similar problem concerning the next terms for $f^{(n)}(0)$. Additionally, since the function is not analytic on zero I thought it could not have a Taylor series around it. Any help will be appreciated.