Asymptotic behavior of reciprocal sum of two products of Fibonacci numbers

[摘要] Let $\{f_{k} \} _{k=1}^{\infty}$ be a Fibonacci sequence with $f_{1}=f_{2}=1$. In this paper, we find a simple form $g_{n}$ such that $$\lim_{n\rightarrow\infty} \Biggl\{ \Biggl(\sum^{\infty}_{k=n}{a_{k}} \Biggr)^{-1}-g_{n} \Biggr\} =0, $$ where $a_{k}=\frac{1}{f_{k}^{2}}$, $\frac{1}{f_{k}f_{k+m}}$, or $\frac{1}{f_{3k}^{2}}$. For example, we show that $$\lim_{n\rightarrow\infty} \Biggl\{ \Biggl(\sum^{\infty}_{k=n}{ \frac {1}{f_{3k}^{2}}} \Biggr)^{-1}- \biggl(f_{3n}^{2}-f_{3n-3}^{2}+ \frac {4}{9}(-1)^{n} \biggr) \Biggr\} =0. $$.

[学科分类] 电力

[关键词] Fibonacci number;Reciprocal sum;Catalan’s identity;Convergent series