Continuous approximation to the reciprocal sum of the cubes of Fibonacci numbers
Abstract
Let fn be the n-th Fibonacci number with f1=f2=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 gn, preserving the principal part fn3−fn−13, 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 gn and these estimates, we completely determine the exact value of the floor function for s=3.
Related
- The infinite sum of the cubes of reciprocal balancing numbers
- Infinite Series About Reciprocal Products of Fibonacci and Lucas Numbers
- Finite-sum Identities and Supercongruences for Cubes of Catalan Numbers
- Space-filling surfaces: sharp Hölder continuous parameterizations from squares to cubes
- An Integral Representation of the Generalized Fibonacci Numbers
BibTeX
@misc{hwang2026continuous,
title = {{Continuous approximation to the reciprocal sum of the cubes of Fibonacci numbers}},
author = {WonTae Hwang and Jond-Do Park and Kyunghwan Song},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18179},
eprint = {2609.18179},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18179},
}
RIS
TY - UNPB
TI - Continuous approximation to the reciprocal sum of the cubes of Fibonacci numbers
AU - WonTae Hwang
AU - Jond-Do Park
AU - Kyunghwan Song
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18179
UR - https://arxiv.org/abs/2609.18179
AB - 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.
ER -