The limit of reciprocal sum of some subsequential Fibonacci numbers - Lee, Ho-Hyeong; Park, Jong-Do (AIMS mathematics, 2021)
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^∞_{k = n}\frac{1}{f_{mk-\ell}} $, where $ m\in\mathbb{N} $ and $ 0≤\ell≤ m-1 $. As a consequence of some inequalities, we prove \begin{document}$ \lim\limits_{n→ ∞}\left\{\left(\sum\limits^∞_{k = n}\frac{1}{f_{mk-\ell}} \right)^{-1} -(f_{mn-\ell}-f_{m(n-1)-\ell})\right\} = 0. $\end{document} And we also compute the explicit value of $ \left\lfloor\left(\sum\limits^∞_{k = n}\frac{1}{f_{mk-\ell}}\right)^{-1}\right\rfloor $. The interesting observation is that the value depends on $ m(n+1)+\ell $.