Does $\sum |a_n|$ Converge?

Question

If $\sum a_n$ converges, then does $\sum |a_n|$ converge as well? This is the same as "absolute convergence", where if $\sum a_n$ converges, then $\sum|a_n|$ might actually diverge, for example, $\sum\frac{(-1)^n}{n}$. So this would serve as a counterexample to the question, and prove it to be wrong, correct?

Answer 1

Yes, you are correct. If $\sum a_n$ converges, $\sum |a_n|$ does not have to converge as well.

Answer 2

Assume a series of positive terms $a_n$. The corresponding alternating series $a_1-a_2+a_3-\cdots$ can be seen as the series with terms $b_1=a_1-a_2,b_2=a_3-a_4,b_3=a_5-a_6,\cdots$ As those terms are differences, they decrease faster than the $a$'s. E.g.

$$a_n=\dfrac1n$$ and $$b_n=\frac1{(2n-1)(2n)}=\Theta\left(\frac1{n^2}\right).$$

For this reason, we can have a series $a_n$ that diverges, while the $b_n$ converge.