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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.

  1. The sequences $\{f_n(z)\}$ and $\{f_n'(z)\}$ converge uniformly on $D$
  2. The series $\sum_{n=1}^\infty f_n$ converge uniformly on $D$
  3. The series $\sum_{n=1}^\infty f_n'$ converge for each $z\in D$
  4. The sequence $\{f_n''(z)\}$ does not converge unless $z=0$

How should i solve this problem?

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What have you tried? Where are you stuck? –  Robert Israel Dec 20 '12 at 8:25
    
Maybe I make a stupid mistake, but $f_n'=f_{n-1}$, right? If so, then in this question there is only one sequence of functions. –  Lior B-S Dec 20 '12 at 10:05

1 Answer 1

Let $R$ be the radius of convergence of $\sum_{n=1}^\infty {z^n\over n}$. Then $1\over R$$=\lim_{n\rightarrow \infty}|{{n+1}\over n}|=1\implies R=1.$ The radius of convergences of $\sum_{n=1}^\infty f_n'$ is thus also $1$ $\implies$ $3$ is correct & $1,2,4$ are false.

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