I wish to expand the function:
$$\dfrac{e^z}{z^n-c^n}$$
about the point $z_0=c$, where c is a constant greater than 0 and n is greater than 2.
So I have that $e^{z-c}$ expands to $1+(z-c)+\frac{1}{2!}(z-c)^2+\frac{1}{3!}(z-c)^3+...$, but I am unsure how to incorporate the denominator. A difference of powers like that could be found using the binomial theorem, but I'm not sure how to generalize this. I have been messing around with it, and I've generalized it to the following:
$$\dfrac{x^n-y^n}{x-n}=x^{n-1}+x^{n-2}y+x^{n-3}y^{2}+...+x^{n-k-1}y^{k}+...+y^{n-1}$$
How can I clean this up? Is this the right method? I need the calculate the integral using the residue theorem, so I need to be able to pull the residue out of this. Thanks!