# Thread: conditionally convergent series - rearrangement

1. does anybody knows an "intuitive" explanation for the fact that you can rearrange a conditionally convergent series in such a way that its infintie sum can be any real number?

have a look at this for an example:
http://www.math.tamu.edu/~tvogel/gallery/node10.html

2.

3. Originally Posted by evariste.galois
does anybody knows an "intuitive" explanation for the fact that you can rearrange a unconditionally convergent series in such a way that its infintie sum can be any real number?

have a look at this for an example:
http://www.math.tamu.edu/~tvogel/gallery/node10.html
It is fairly simple.

First you note that the positive and negative numbers in the series taken separately form series that diverge. This must be so since the original series does not converge absolutely. Not also that they converge to zero as sequances, since the series is summable.

Now arrange the positive and negative numbers in decreasing order of absolute value..

Then pick a number that you want to be the limit of the re-arranged series. For simplicity, let's assume it is positive. Call it L.

Select numbers from the positive list, in order (remember that they are in decreasing order of magnitude) until the first time that the sum is greater than L. Next select negative numbers from the list, in order, until the revised sum is less than L. Now return to the positive numbers. Repeat. The resulting series, defined inductively in this manner converges to L.

If you can't turn this into a rigorous proof yourself, I think there is a proof in an old calculus book by Tom Apostol that is reasonably clear. I don't recall the precise title.

4. its really quite simple if you think about it that way. thanks a lot for your help!

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