Integer Ratios of Some Consecutive Series

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Sesthawuth Petchkaew
Supawadee Prugsapitak

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\}).

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How to Cite
Petchkaew, S., & Prugsapitak, S. (2021). Integer Ratios of Some Consecutive Series. Mathematical Journal, 66(705), 25–31. Retrieved from https://ph02.tci-thaijo.org/index.php/MJMATh/article/view/242459
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.