To show that $\displaystyle \sum_n^{\infty} \frac{z^n}{1-z^n}$ converges uniformly on compact subsets of D(0,1) (the unit disc centered at the origin), does is suffice to show that the sum converges uniformly on an arbitrary closed subdisc?

Subsequently, I must find the power series coefficients. How do I do this? Does anyone have general guidelines or a worked example I could analogize?

Muchas gracias