Let $D=\{z\in\mathbb{C}\mid |z|<1\}$ and let $f_n:D\to \mathbb{C}$ be defined by $f_n(z)=\frac{z^n}{n}$ for $n=1,2,\ldots$ Then which of the followings are true.
- The sequences $\{f_n(z)\}$ and $\{f_n'(z)\}$ converge uniformly on $D$
- The series $\sum_{n=1}^\infty f_n$ converge uniformly on $D$
- The series $\sum_{n=1}^\infty f_n'$ converge for each $z\in D$
- The sequence $\{f_n''(z)\}$ does not converge unless $z=0$
How should i solve this problem?