TITLE: Prove that $\sum_{n=0}^{\infty}{{a_n}{z^n}}$ converges absolutely and uniformly in $D$.
QUESTION [8 upvotes]: PROBLEM

Suppose that the complex series $\displaystyle \sum_{n=0}^{\infty}{a_n}$ converges.  Let $r < 1$ and set $D = \{z \in \mathbb{C} : |z| < r\}$.

Prove that $\displaystyle \sum_{n=0}^{\infty}{{a_n}{z^n}}$ converges absolutely and uniformly in $D$.


MY FUTILE ATTEMPTS
I know that I must somehow use the result(s) in this question: If the complex series $\displaystyle \sum_{n=0}^{\infty}{a_n}$ converges, show that there exists a positive number $A$ such that $|a_n| \leq A$ for all $n$..
Also, the easiest way that I can think of to solve this problem is to use the following theorem:
THEOREM (Weierstrass M-Test or Dominated Convergence Theorem)


Given the series of functions $\ \displaystyle \sum_{n=1}^{\infty}{{f_n}(z)}, z \in E$.  Suppose that $\{M_n\}$ is a sequence of positive real numbers such that
    (i) $|{f_n}(z)| \leq M_n, \forall n \in \mathbb{N}, \forall z \in E$.  (ii) $\displaystyle \sum_{n=1}^{\infty}{M_n}$ converges.  Then $\displaystyle \sum_{n=1}^{\infty}{{f_n}(z)}$ converges absolutely and uniformly on $E$.


Of course I know that I need to set ${f_n}(z) = {a_n}{z^n}$.  Then ${f_n}(z)$ is a power series.  What I don't know is the sequence $\{M_n\}$.
QUESTIONS
(1) Is the Weierstrass M-Test indeed the best way to tackle this problem?  If so, what should be my $M_n$?
(2) If the answer to the first question in (1) is NO, how can I be able to solve this problem?

REPLY [2 votes]: We use the Weierstrass $M$-Test.  
As hinted in the comments, we just need to pick ${f_n}(z) = {a_n}{z^n}$ and $M_n = Ar^n$, where $A$ is a bound for $\{a_n\}$ (by the result in If the complex series $\displaystyle \sum_{n=0}^{\infty}{a_n}$ converges, show that there exists a positive number $A$ such that $|a_n| \leq A$ for all $n$.).  Note that
(i) $\left|{f_n}(z)\right|=\left|{a_n}{z^n}\right|=|a_n||z^n| \leq A{r^n} = M_n, \forall n \in \mathbb{N}, \forall z \in D$.
(ii) $\sum_{n=0}^{\infty}{M_n}$ converges by the Ratio Test.
Therefore, $\sum_{n=0}^{\infty}{{a_n}{z^n}}$ converges absolutely and uniformly in $D$.