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Evaluating the Svein-Graham Sum

  1. Apr 28, 2015 #1
    Good evening dearest physicians and mathematicians,

    I recently came across the so-called "Svein-Graham sum", and i wondered: is it possible to find a simple formula for evaluating it?
    [itex]\sum_{i=0}^k x\uparrow\uparrow i = \left .1+x+x^x+x^{x^x}+ ... +x^{x^{x^{x^{.^{.^{.^x}}}}}}\right \}k[/itex]
  2. jcsd
  3. Apr 29, 2015 #2
    Hi, I use Mathematica to define a function sg[x,k] to calculate the Svein-Graham sum and plot some figures for ##x \in [1,2]## with ##k## varies from 1 to 5.
    Code (Text):
    sg[x_, k_] := Module[{f},
      f[y_] := #^y &;
      (FoldList[f[x], x, Range[k - 1]] // Total) + 1]
    Plot[sg[x, #], {x, 1, 2}] & /@ Range[1, 5]
    Last edited: Apr 29, 2015
  4. Apr 29, 2015 #3
    I was looking for a more analytic expression like [itex]\sum_{i=1}^n i = \frac{n(n+1)}{2}[/itex]. Maybe it's possible to find yet another connection to the Bernoulli numbers? But thank you nevertheless!
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