Hello there,

My friends and I were talking and we have been wondering the following:

If the absolute value of a series converges, does the series converge?

This came up when we were talking about alternating series'.

Thanks,

Rob

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- Apr 11th 2013, 01:35 PMRobertXIVRandom Series Question
Hello there,

My friends and I were talking and we have been wondering the following:

If the absolute value of a series converges, does the series converge?

This came up when we were talking about alternating series'.

Thanks,

Rob - Apr 11th 2013, 01:55 PMPlatoRe: Random Series Question
If this is what you mean: If $\displaystyle \sum\limits_{n = K}^\infty {\left| {{a_n}} \right|} $

**converges**then $\displaystyle \sum\limits_{n = K}^\infty { {{a_n}} } $ also converges.

Yes that is a theorem.

But you posted " the absolute value of a series converges". That makes no sense.

Which is it? - Apr 11th 2013, 03:14 PMRobertXIVRe: Random Series Question
Sorry, sorry, I meant the first one. Thanks a lot!

- Apr 11th 2013, 03:43 PMPlatoRe: Random Series Question

The first one is the series of absolute values whereas what you posted is the absolute value of a series. You can see the difference.

Here why it works.**Series convergence is**__all about__the convergence of a sequence of partial sums.

See that $\displaystyle 0\le \left| {\sum\limits_{n = 1}^K {{a_n}} } \right| \leqslant \sum\limits_{n = 1}^K {\left| {{a_n}} \right|} $.

Thus if the sequence $\displaystyle { \sum\limits_{n = 1}^K {\left| {{a_n}} \right|}$ converges so does $\displaystyle {\sum\limits_{n = 1}^K {{a_n}} $