TITLE: $\sum a_n $ is convergent and $|x|<1$ then, is $\sum a_n x^n$ absolutely convergent?
QUESTION [0 upvotes]: It is clear when, $a_n>0$ 
$|x^n| <1$ and $|a_n x^n| < |a_n| = a_n$ 
Since $\sum a_n$ is convergent by comparison test $\sum |a_n x^n|$ is convergent
But will it converge absolutely even if $\sum a_n$ is not a series of postive term?

REPLY [2 votes]: Since $\sum a_n$ converges, we have $\lim_{n\to\infty} a_n = 0$, hence there is a positive integer $N$ such that $|a_n| \le 1$, for all $n \ge N$.

Then for $n \ge N$, we get $|a_nx^n| = |a_n||x|^n \le |x|^n$.

Hence by the comparison test, since $\sum_{n=N}^\infty |x|^n$ converges, we get that $\sum_{n=N}^\infty |a_nx^n|$ converges.

It follows that $\sum a_nx^n$ converges absolutely.
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