| Autor: |
Ho-Hyeong Lee, Jong-Do Park |
| Jazyk: |
angličtina |
| Rok vydání: |
2021 |
| Předmět: |
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| Zdroj: |
AIMS Mathematics, Vol 6, Iss 11, Pp 12379-12394 (2021) |
| Druh dokumentu: |
article |
| ISSN: |
2473-6988 |
| DOI: |
10.3934/math.2021716?viewType=HTML |
| Popis: |
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 $. |
| Databáze: |
Directory of Open Access Journals |
| Externí odkaz: |
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