Given the sum: $$ \sum_{i = 0}^n i^{1/5} $$
How to find $A$ in: $$ \sum_{i = 0}^n i^{1/5} = A + O(\frac1{n^6}) $$ I tried to use Euler–Maclaurin formula and obtained numbers that confused me?
2025-01-12 23:32:13.1736724733
How to find the sum: $ \sum_{i = 0}^n i^{1/5} $
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With the help of Mathematica: $$ \sum_{j=1}^k j^{1/5}= \frac{5k^{6/5}}{6}+\frac{k^{1/5}}{2}-\frac{(\sqrt{5}-1)\Gamma(6/5)\zeta(6/5)}{\sqrt[5]{64\pi^6}}+\frac{1}{60}k^{-4/5}-\frac{1}{2500}k^{-14/5}+ $$ $$ +\frac{19}{187500}k^{-24/5}-\frac{551}{7812500}k^{-34/5}+\mathcal{O}\left(k^{-44/5}\right) $$