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'.
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 .
Thus if the sequence converges so does