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I Summation of 1^1+2^2+3^3+...+k^k

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  1. Jun 8, 2017 #1
    Does that summatiom have a shorter representation at all?
    ##\sum_{n=1}^{k} n^n = ?##
    I guess it is not of the form of constant power series, but I could not find an alternative.

    Mentor note: made formula render properly
     
    Last edited by a moderator: Jun 8, 2017
  2. jcsd
  3. Jun 8, 2017 #2

    mathman

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  4. Jun 8, 2017 #3

    mfb

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    Huh? Looks both like nn to me.

    I'm not aware of an analytic expression. It can probably be approximated with the Stirling formula and then some integration.
     
  5. Jun 8, 2017 #4
    It's kk :-)
     
  6. Jun 8, 2017 #5

    fresh_42

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    Maybe one can use Faulhaber to rewrite ##n^n## as difference of ##\sum_{k=1}^n k^n - \sum_{k=1}^{n-1} k^n## to get an expression in Bernoulli numbers which can then be summed again. A giant polynomial of Bernoulli numbers. Of course my bet to the original question
    is NO. I mean the length of the expression is seven! Almost impossible to shorten.
     
  7. Jun 8, 2017 #6

    berkeman

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    Not true for an engineer for k>5 or so...

    k^k
     
  8. Jun 9, 2017 #7

    fresh_42

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    Now as you say it. Mathematicians can also shorter ...
    ##O(1)##
     
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