Complex Analysis (expand 1/z in a power series)

[Guest]

I have a few questions that I am unsure of how to approach:

For the function \(\displaystyle \frac{1}{z},\, c\,=\,i\), please expand it in a power series about the respective center and determine the radius and the circle of convergence

Thanks for any and all help

 

[pka]

I will get you started.
\(\displaystyle \L
\frac{1}{i} = - i\quad \Rightarrow \quad \frac{1}{{\left( i \right)^n }} = \left( { - i} \right)^n\)

If \(\displaystyle \L
f(z) = \frac{1}{z}\quad \Rightarrow \quad f^{(n)} (z) = \frac{{\left( { - 1} \right)^n \left( {n + 1} \right)!}}{{z^{n + 1} }}\).

Therefore, \(\displaystyle \L
f^{(n)} (i) = \left( { - 1} \right)^n \left( {n + 1} \right)!\left( { - i} \right)^{n + 1}\).