Integer Ratios of Some Consecutive Series

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Published: Dec 18, 2021
Keywords:
Series, Consecutive series

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Sesthawuth Petchkaew
Algebra and Applications Research Unit, Division of Computational Science, Faculty of Science, Prince of Songkla University, Songkhla 90110
Supawadee Prugsapitak
Algebra and Applications Research Unit, Division of Computational Science, Faculty of Science, Prince of Songkla University, Songkhla 90110

Abstract

In this article, we establish necessary and sufficient conditions for the ratio of some consecutive series to be integers. The series that we investigate in the article are as follows:


gif.latex?B_k&space;(n)=\frac{1}{1(1+k)}+\frac{1}{2(2+k)}+\cdots+\frac{1}{n(n+k)}~~(n,k\in\mathbb{N})


gif.latex?C_{a,d}(n)=\sum_{i=1}^n&space;a+(n-i)d~~(a,d,n\in&space;\mathbb{N})


and


gif.latex?D_{a,r}(n)=a+ar+\cdots+ar^{n-1}~~(a,n\in\mathbb{N},&space;r\in&space;\mathbb{Q}\setminus&space;\{0\}).

Article Details

How to Cite
Petchkaew, S., & Prugsapitak, S. (2021). Integer Ratios of Some Consecutive Series. Mathematical Journal by The Mathematical Association of Thailand Under The Patronage of His Majesty The King, 66(705), 25–31. Retrieved from https://ph02.tci-thaijo.org/index.php/MJMATh/article/view/242459
Issue
Vol. 66 No. 705: September – December 2021
Section
Academic Article

References

Baoulina, I. N., Moree, P. (2016). Forbidden Integer Ratios of Consecutive Power Sums. In: Sander, J., Steuding, J., Steuding, R., eds. From Arithmetic to Zeta-Functions: Number Theory in Memory of Wolfgang Schwarz, p. 1 - 30. Springer.

Baoulina, I. N. (2019). Integer Ratios of Consecutive Alternating Power Sums. The American Mathematical Monthly, 126 (7), p. 651 - 654.

Moree, P. (2013). Moser’s Mathemagical Work on The Equation 1^k + 2^k + cdots + (m-1)^k = m^k. Rocky Mounta in J. Math. 43 (5), p. 1707 - 1737.