Continuous approximation to the reciprocal sum of the cubes of Fibonacci numbers

RESUMO

Let $f_n$ be the $n$-th Fibonacci number with $f_1=f_2=1$. Recently exact formulas for the integer parts of the tails of inverse reciprocal Fibonacci numbers have been obtained in several cases. However, the cubic case ($s=3$) is much more complicated because of highly oscillating error terms. Thus it is difficult to construct a precise continuous approximation and algebraic estimates simultaneously. In this paper, we give a complete and unified algebraic method to solve this difficulty. More precisely we construct an explicit closed form of sequence $g_n$, preserving the principal part $f_n^3-f_{n-1}^3$, such that $\ds \lim_{n\rightarrow\infty}\left\{ \left( \sum^\infty_{k=n}\frac{1}{f_k^3} \right)^{-1}-g_n \right\}=0. $ By decomposing the error terms and investigating the algebraic identities, we establish the lower and upper bounds $ \ds g_n<\left( \sum^\infty_{k=n}\frac{1}{f_k^3} \right)^{-1}<g_n+2/f_n $ for sufficiently large $n$. As an application of the explicit form of the sequence $g_n$ and these estimates, we completely determine the exact value of the floor function for $s=3$.