# If $\sum \limits_{k=1}^\infty a_k$ converges absolutely, does it imply that $\sum \limits_{k=1}^\infty |a_k|$ converge? [closed]

I understand that the following statement is true: If $\sum \limits_{k=1}^\infty |a_k|$ converges, then $\sum \limits_{k=1}^\infty a_k$ converge absolutely.

So I'm just wondering if the following is true as well: If $\sum \limits_{k=1}^\infty a_k$ converges absolutely, then $\sum \limits_{k=1}^\infty |a_k|$ converge.

## closed as unclear what you're asking by Did, ncmathsadist, user26857, pjs36, ShaileshFeb 26 '16 at 0:02

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• This is true by definition of "absolutely convergence". – Crostul Feb 25 '16 at 16:48
• This is literally the definition. – M10687 Feb 25 '16 at 16:51
• In English usage "absolutely" means really, or truly, or precisely, or definitely. But, as indicated here, "absolutely convergent" simply means that the series with absolute values inserted converges. Fortunately for our intuition when a series converges absolutely it really and truly does converge too. It is easy to lose sight of the actual definitions of the terms we employ and then ask questions like this. We all have done it. – B. S. Thomson Feb 25 '16 at 17:13

## 2 Answers

That's exactly what "absolute convergence means": the series $\sum_ka_k$ converges absolutely if $\sum_k|a_k|$ converges.

In definitions such as $$\mbox{Definition: \sum a_k converges absolutely if \sum|a_k| converges},$$ the word "if" should be read as "is a synonym for". This might be the source of your confusion.