I have been trying to apply the ratio test onto $\dfrac{z^n}{1+z^n}$. After the usual initial steps.

I need to show that $$\lim\_{n \to \infty} \left|\dfrac{z(1+z^n)}{1+z^{n+1}}\right|<1$$ 

I am unsure of how to make further progress, and so what is the trick from here? 

Once this is shown, we can therefore say that the series
 $$\displaystyle\sum\limits^\infty\_{n=0} \dfrac{z^n}{1+z^n}$$
converges absolutely. 

Before I can deduce $\displaystyle\sum\limits^\infty\_{n=0} \dfrac{z^n}{1+z^n}$ is holomorphic on $D=\{z \in \mathbb{C} \mid |z| <1\},$ would I have to show uniform convergence? If so, what is the simplest way?