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How can I calculate the value of the infinite sum $$\sum_{x=1}^\infty x^{-x}$$ By comparison to a geometric series, I know that it converges, but I don't know how to calculate an exact value. So far I have tried the following:

  • I have tried to turn it into a telescoping sum, but have had no luck

  • I tried adding a variable and differentiating it to create an easier sum

Any hints?

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marked as duplicate by Simply Beautiful Art, Community Jun 9 '17 at 20:25

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    $\begingroup$ The sophomore's dream. But apparently the actual value can't be written in "closed form". $\endgroup$ – Robert Israel Jun 9 '17 at 20:10
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    $\begingroup$ It converges so fast that its actual value, for practical purposes, is just the sum of its first five terms or so. $\endgroup$ – Jack D'Aurizio Jun 9 '17 at 20:12
  • $\begingroup$ See also OEIS sequence A073009. $\endgroup$ – Robert Israel Jun 9 '17 at 20:13