The limit of reciprocal sum of some subsequential Fibonacci numbers

• Published in: AIMS mathematics Vol. 6; no. 11; pp. 12379 - 12394
• Main Authors: Lee, Ho-Hyeong, Park, Jong-Do
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2021
• Subjects: catalan's identity; convergent series; fibonacci number; floor function; reciprocal sum
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2021716

This paper deals with the sum of reciprocal Fibonacci numbers. Let $ f_0 = 0 $, $ f_1 = 1 $ and $ f_{n+1} = f_n+f_{n-1} $ for any $ n\in\mathbb{N} $. In this paper, we prove new estimates on $ \sum\limits^\infty_{k = n}\frac{1}{f_{mk-\ell}} $, where $ m\in\mathbb{N} $ and $ 0\leq\ell\leq m-1 $. As a consequence of some inequalities, we prove <disp-formula> <tex-math id="FE1"> \begin{document}$ \lim\limits_{n\rightarrow \infty}\left\{\left(\sum\limits^\infty_{k = n}\frac{1}{f_{mk-\ell}} \right)^{-1} -(f_{mn-\ell}-f_{m(n-1)-\ell})\right\} = 0. $\end{document} </tex-math></disp-formula> And we also compute the explicit value of $ \left\lfloor\left(\sum\limits^\infty_{k = n}\frac{1}{f_{mk-\ell}}\right)^{-1}\right\rfloor $. The interesting observation is that the value depends on $ m(n+1)+\ell $.