I've no clue whether my approach leads to a correct result, but I'd first try with some very dirty tricks (beware!!):
Differentiate the complex series with respect to and you get
Now I'll just take the identity valid for and hope it converges for most complex numbers on the border of the closed disk (needs checking!). I get
Now what's the logarithm of a complex number? For a complex number , I have
whence . Hence
You can work out the rest. Take the real part, and try to find the antiderivative. Check by numerical simulation, before trying to rigorize all the above.
I'd also be interested in a rigorous (and maybe more elegant?) derivation. Thanks to anybody who can provide it!