1. ## Random 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

2. ## Re: Random Series Question

Originally Posted by RobertXIV
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'
If this is what you mean: If $\sum\limits_{n = K}^\infty {\left| {{a_n}} \right|}$ converges then $\sum\limits_{n = K}^\infty { {{a_n}} }$ also converges.
Yes that is a theorem.

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

Which is it?

3. ## Re: Random Series Question

Sorry, sorry, I meant the first one. Thanks a lot!

4. ## Re: Random Series Question

Originally Posted by RobertXIV
Sorry, sorry, I meant the first one. Thanks a lot!

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

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