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

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 $\text{ds}{\lim }_{n\to \mathrm{\infty }}\{{\left({\sum }_{k=n}^{\mathrm{\infty }}\frac{1}{f_k^3}\right)}^{-1}-g_n\}=0.$ By decomposing the error terms and investigating the algebraic identities, we establish the lower and upper bounds $\text{ds}{g}_n<{\left({\sum }_{k=n}^{\mathrm{\infty }}\frac{1}{f_k^3}\right)}^{-1}<g_n+2\mathrm{/}{f}_n$ 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 $s=3$.