Getting stuck on series - can't develop Taylor series.

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.

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= - 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.