In the paper, we obtain asymptotic expansion of the finite sum of some sequences $ S_{n} = \sum_{k = 1}^{n}\left(n^{2}+k\right) ^{-1} $ by using the Euler's standard one of the harmonic numbers.
Citation: Ling Zhu. Asymptotic expansion of a finite sum involving harmonic numbers[J]. AIMS Mathematics, 2021, 6(3): 2756-2763. doi: 10.3934/math.2021168
In the paper, we obtain asymptotic expansion of the finite sum of some sequences $ S_{n} = \sum_{k = 1}^{n}\left(n^{2}+k\right) ^{-1} $ by using the Euler's standard one of the harmonic numbers.
| [1] | R. L. Graham, D. E. Knuth, O. Patashnik, Concrete mathematics: A foundation for computer science, 2 Eds., Amsterdam: Addison-Wesley Publishing Company, 1994. |
| [2] | D. E. Knuth, Euler's constant to 1271 places, Math. Comput., ${\bf 16}$ (1962), 275–280. |
| [3] | A. Jeffrey, Handbook of mathematical formulas and integrals, 3 Eds., Elsevier Academic Press, 2004. |
Ling Zhu. Asymptotic expansion of a finite sum involving harmonic numbers[J]. AIMS Mathematics, 2021, 6(3): 2756-2763. doi: 10.3934/math.2021168
DownLoad: