Asymptotic behavior of reciprocal sum of two products of Fibonacci numbers

Abstract

AbstractLet $\{f_{k} \} _{k=1}^{\infty}${fk}k=1∞ be a Fibonacci sequence with $f_{1}=f_{2}=1$f1=f2=1. In this paper, we find a simple form $g_{n}$gn such that $$\lim_{n\rightarrow\infty} \Biggl\{ \Biggl(\sum^{\infty}_{k=n}{a_{k}} \Biggr)^{-1}-g_{n} \Biggr\} =0, $$limn→∞{(∑k=n∞ak)−1−gn}=0, where $a_{k}=\frac{1}{f_{k}^{2}}$ak=1fk2, $\frac{1}{f_{k}f_{k+m}}$1fkfk+m, or $\frac{1}{f_{3k}^{2}}$1f3k2. For example, we show that $$\lim_{n\rightarrow\infty} \Biggl\{ \Biggl(\sum^{\infty}_{k=n}{ \frac {1}{f_{3k}^{2}}} \Biggr)^{-1}- \biggl(f_{3n}^{2}-f_{3n-3}^{2}+ \frac {4}{9}(-1)^{n} \biggr) \Biggr\} =0. $$limn→∞{(∑k=n∞1f3k2)−1−(f3n2−f3n−32+49(−1)n)}=0.