The proof of a formula concerning the asymptotic behavior of the reciprocal sum of the square of multiple-angle Fibonacci numbers.

Let (F n) n be the Fibonacci sequence defined by F n + 2 = F n + 1 + F n with F 0 = 0 and F 1 = 1 . In this paper, we prove that for any integer m ≥ 1 there exists a positive constant C m for which lim n → ∞ { (∑ k = n ∞ 1 F m k 2) − 1 − (F m n 2 − F m (n − 1) 2 + (− 1) m n C m) } = 0. Furthermore, we show that C m tends to 2 / 5 as m → ∞ (indeed, we provide quantitative versions of the previous results as well as an explicit form for C m ). This confirms some questions proposed by Lee and Park [J. Inequal. Appl. 2020(1):91 2020]. [ABSTRACT FROM AUTHOR]