Is there a closed form for the following sum?
$$\sum_{n=0}^\infty e^{-\sqrt n}$$
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2Wolfram Alpha does not show an expression, so probably there is no closed form. – Peter Jul 18 '16 at 15:31
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Various inverse symbolic calculators come up with nothing either. – George V. Williams Jul 18 '16 at 17:04
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Depends upon the meaning of "closed form"; for instance, one could probably add/subtract cosh(), sinh() and do some symmetry touch-up. – rrogers Jul 19 '16 at 21:15
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1$$\int_{0}^{+\infty}e^{-\sqrt{x}},dx = 2$$ and the difference between the sum and the above integral can be estimated through Abel's or Euler-MacLaurin summation formulas. – Jack D'Aurizio Sep 14 '16 at 23:45
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1Based on my answer to 1, we can write the answer as a more interesting sum $$ \sum_{n=0}^\infty e^{-\sqrt{n}}=\frac{5}{2} - \sum_{s=1}^\infty \frac{\Gamma(1+\frac{s}{2})\zeta(1+\frac{s}{2})\sin(\frac{\pi s}{4})}{(-1)^s \pi(2\pi)^{s/2} \Gamma(s+1)} $$ – Benedict W. J. Irwin Jun 29 '17 at 17:26
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Also related. – metamorphy Apr 04 '23 at 08:03