The asymptotic behavior of the reciprocal sum of generalized Fibonacci numbers

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Abstract

Let $\left(u_{n}\right)_{n \geq 0}$ be the special Lucas u-sequence defined by

$u_{n+2}=A u_{n+1}-B u_{n}, \quad u_{0}=0, u_{1}=1,$

where n ≥ 0, B = ±1, and A is an integer such that A² − 4B > 0. Let

$a_{k}=\frac{1}{u_{m k}^{s}}, \frac{1}{u_{m k}+u_{m k+l}}, \frac{1}{\sum_{i=0}^{l} u_{m k+i}}, \frac{1}{u_{m k} u_{m k+2 l}}, \frac{1}{u_{m k} u_{m k+2 l-1}}, \frac{1}{u_{m k}+C},$

where m, l are positive integers, s = 1, 2, 3, 4, and C is any constant. The aim of this paper is to find a form g_n such that

$\lim _{n \rightarrow \infty}\left(\left(\sum_{k=n}^{\infty} a_{k}\right)^{-1}-g_{n}\right)=0 .$

For example, we show that

$\lim _{n \rightarrow \infty}\left(\left(\sum_{k=n}^{\infty} \frac{1}{u_{m k}}\right)^{-1}-\left(u_{m n}-u_{m(n-1)}\right)\right)=0 .$

Keywords

generalized Fibonacci number / reciprocal sum / asymptotic formulas

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