Fibonacci数立方倒数和连续逼近
Continuous approximation to the reciprocal sum of the cubes of Fibonacci numbers
- Jeonbuk National University（全北国立大学）
- Kyung Hee University（庆熙大学）
- Jeju National University（济州国立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对Fibonacci数立方倒数和尾部，构造显式闭式序列$g_n$，建立其与倒数和的误差极限及上下界，从而完全确定$s=3$时取整函数的精确值。
AI中文摘要：
设$f_n$为第$n$个Fibonacci数，其中$f_1=f_2=1$。最近，在若干情形下已获得倒Fibonacci数尾部和的整数部分的精确公式。然而，三次情形（$s=3$）因误差项高度振荡而复杂得多，因此难以同时构造精确的连续逼近和代数估计。本文给出一种完整且统一的代数方法来解决这一困难。更精确地，我们构造了序列$g_n$的显式闭式形式，保留主部$f_n^3-f_{n-1}^3$，使得$\ds \lim_{n\rightarrow\infty}\left\{ \left( \sum^\infty_{k=n}\frac{1}{f_k^3} \right)^{-1}-g_n \right\}=0.$ 通过分解误差项并研究代数恒等式，我们建立了下界和上界$\ds g_n<\left( \sum^\infty_{k=n}\frac{1}{f_k^3} \right)^{-1}<g_n+2/f_n$对充分大的$n$成立。作为序列$g_n$显式形式及这些估计的应用，我们完全确定了$s=3$时取整函数的精确值。
英文摘要：
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 $s=3$.
