The limit of reciprocal sum of some subsequential Fibonacci numbers

Abstract

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 $ \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. $ 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 $.

Keywords

• fibonacci number
• reciprocal sum
• floor function
• convergent series
• catalan's identity