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
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?
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}} $