Abstract
In this article, we obtain two interesting families of partial finite sums of the reciprocals of the Fibonacci numbers, which substantially improve two recent results involving the reciprocal Fibonacci numbers. In addition, we present an alternative and elementary proof of a result of Wu and Wang.
Highlights
The Fibonacci sequence [ ], Sequence A is defined by the linear recurrence relationFn = Fn– + Fn– for n ≥, where Fn is the nth Fibonacci number with F = and F =
The Fibonacci sequence plays an important role in the theory and applications of mathematics, and its various properties have been investigated by many authors; see [ – ]
In [ ], the partial infinite sums of the reciprocal Fibonacci numbers were studied by Ohtsuka and Nakamura
Summary
The Fibonacci sequence [ ], Sequence A is defined by the linear recurrence relation. There has been an increasing interest in studying the reciprocal sums of the Fibonacci numbers. In [ ], the partial infinite sums of the reciprocal Fibonacci numbers were studied by Ohtsuka and Nakamura. They established the following results, where · denotes the floor function. Wu and Wang [ ] studied the partial finite sum of the reciprocal Fibonacci numbers and deduced that, for all n ≥ ,. Inspired by Wu and Wang’s work, we obtain two families of partial finite sums of the reciprocal Fibonacci numbers in this paper, which significantly improve Ohtsuka and Nakamura’s results, Theorems .
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