# Does $\sum |a_n|$ Converge?

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?

• That looks correct Mar 23, 2021 at 20:46

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

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.

• (+1) I like that this goes further than reiterating the question.
– robjohn
Mar 23, 2021 at 21:54
• @robjohn: yep, I tried to give the OP some intuition on why it is so.
– user65203
Mar 23, 2021 at 22:53