Is the series $\sum_{n=1}^\infty \dfrac {z^n}{1+z^n}$ of complex numbers uniformly convergent in $B(0,1)$?

Question

Is the series $\sum_{n=1}^\infty \dfrac {z^n}{1+z^n}$ of complex numbers uniformly convergent in $B(0,1)$ ? I know it is point-wise convergent in $B(0,1)$ , I also know that it is uniformly convergent in $B[0;r]$ for every $0<r<1$ , but can't figure out about the open domain . Please help . Thanks in advance

Answer 1

Hint: If $\sum f_n$ converges uniformly on a set $E,$ then $f_n$ converges uniformly to $0$ on $E.$ Is that true in this case?