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?

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    $\begingroup$ That looks correct $\endgroup$ Mar 23, 2021 at 20:46

2 Answers 2


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.

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

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