Hi, I have read this Sum of 1.5-powers of natural numbers and answers posted by Mark Viola. However, I am not too sure on how he got to this. Can anyone show me as I am quite new to series?
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Maximus Su
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Did you follow the link to the Euler-Maclaurin formula? – Gerry Myerson Jun 29 '21 at 07:16
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2The formula in the linked answer is off at the constant term (see Claude Leibovici's answer for correct expansion). In general, Euler-Maclaurin's formula is great for those parts which diverges at large $n$. For constant term and beyond, Abel-Plana formula or Mellin's transform will be a better tool. – achille hui Jun 29 '21 at 08:52
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It is quite simple using the generalized harmonic numbers $$S_n=\sum_{k=1}^n k^{\frac 32}=H_n^{\left(-\frac{3}{2}\right)}$$ Using their asymptotics for large values of $n$, you have $$S_n=\frac25n^{5/2}+\frac12 n^{3/2}+\frac18n^{1/2}+\zeta \left(-\frac{3}{2}\right)+\frac{1}{1920}n^{-3/2}-\frac {1 }{21504}n^{-5/2}+O\left({n^{-7/2}}\right)$$ $$\zeta \left(-\frac{3}{2}\right)\sim -0.0254852$$
Claude Leibovici
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