Notices
Results 1 to 3 of 3

Thread: conditionally convergent series - rearrangement

  1. #1 conditionally convergent series - rearrangement 
    Forum Freshman
    Join Date
    Oct 2007
    Posts
    39
    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


    Reply With Quote  
     

  2.  
     

  3. #2 Re: Unconditionally convergent series - rearrangement 
    . DrRocket's Avatar
    Join Date
    Aug 2008
    Posts
    5,486
    Quote 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.


    Reply With Quote  
     

  4. #3  
    Forum Freshman
    Join Date
    Oct 2007
    Posts
    39
    its really quite simple if you think about it that way. thanks a lot for your help!
    Reply With Quote  
     

Bookmarks
Bookmarks
Posting Permissions
  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •