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Rearrangement of infinite series

  1. Nov 5, 2005 #1
    I'm working on this problem:
    Prove that if |x|<1, then
    1 + x^2 + x + x^4 + x^6 + x^3 + x^10 + x^5 + ...= 1/(1-x).
    I know that this is a rearrangement of the (absolutely convergent) geometric series, so it converges to the same limit. My trouble is proving that the rearrangement represents a 1-1 and onto mapping of the natural number onto themselves. I wrote out the terms and found the pattern, but I can't seem to get a formula for it. I'm stuck. Here's what I got:The first column represents the geometric series. The number after the colon is the rearrangement, and the last column is how I related the rearrangement to the origional goemetric series.

    It has a definite pattern, and I can see it, but I can't write a formula down that works so that I can show it's bijective. I'm not sure that I am going about this correctly.
    Any input will be appreciated.
  2. jcsd
  3. Nov 5, 2005 #2


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    Hint, perhaps.

    How do you construct a bijection from [tex]\mathbb{N}[/tex] to [tex]\mathbb{Z}[/tex]?
  4. Nov 5, 2005 #3
    I let 0=1, -n=n+1 and n=2n+1...or something similar.
    How does that help me map [tex]\mathbb{N}[/tex] to [tex]\mathbb{N}[/tex] in my situation?
  5. Nov 5, 2005 #4
    any help? Any thoughts?
  6. Nov 5, 2005 #5


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    Seems pretty simple to me, for n, some natrual number.

    3n ==> 4n
    3n + 1 ==> 4n + 2
    3n + 2 ==> 2n + 1

    Doesn't take much beyond that to work it out.
  7. Nov 6, 2005 #6
    thanks! I just had a block on getting the formula for the pattern. I now have it solved! AWESOME!!
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