Asymptotic behavior of reciprocal sum of two products of Fibonacci numbers

Abstract

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

Keywords

• Fibonacci number
• Reciprocal sum
• Catalan’s identity
• Convergent series