Are you just trying to find the first few terms of the expansion, or also the general coefficient? Because this would be much more complicated. The cosecant has a mean-looking series expansion involving the Bernoulli numbers.
Find convergences radius at z = 1 and develop a Taylor series.
Just by looking at the function I see that deviser and numerator has zeros in z = 1, so theres a limit in z = 1. The numerator isn't defined in z = 0. So the convergence radius is |z-1| < 1. (See deviser also, zeros in z = 2, 3, 4 ...)
First the log function, I think I've got it right. The Taylor expansion is also pretty straight forward.
= = =
In this part I'm starting to get shaky.
= - This is where I need some pointers. I can't simplify this enough to multiply it with the log z part to get the whole series.