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Proof that zeta(-1) is -1/12

  1. Jan 17, 2014 #1

    bgq

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    Hi,
    I have read that zeta(x) = 1^(-x) + 2^(-x) + 3^(-x) + ... infinity
    for x = -1, zeta(-1) = 1 + 2 + 3 + 4 + 5 ...
    What confused me is that zeta(-1) = -1/12 and so 1 + 2 + 3 + 4 + 5 + ... = -1/12
    Can anybody give a proof that zeta(-1) is -1/12.

    Thanks in advance.
     
  2. jcsd
  3. Jan 17, 2014 #2

    Office_Shredder

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    We have a fairly substantial thread on this topic already

    https://www.physicsforums.com/showthread.php?t=732197

    The main gist is that [itex] \zeta(x) [/itex] is a function such that if x>1,
    [tex] \zeta(x) = 1^{-x} + 2^{-x} +.... [/tex]
    You can extend this function to allow for values of x smaller than 1, but when you do you aren't really calculating the sum of natural numbers anymore. In particular trying to do manipulations as if you really are calculating that sum can be dangerous. See the thread for more details and feel free to continue the discussion there.
     
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