The inverse of the (alternating) infinite sum of the reciprocal of the weighted sum for generalized Fibonacci sub-sequences

Abstract

In this paper, for the generalized Fibonacci sequence $\left\{W_n\left(a,b,p,q\right)\right\}$, by using elementary methods and techniques, we give the asymptotic estimation values of ${\left(\sum _{k=n}^{\infty }\frac{1}{\sum _{i=0}^t{s}_i{W}_{mk+l_i}}\right)}^{-1}$ and ${\left(\sum _{k=n}^{\infty }\frac{{\left(-1\right)}^k}{\sum _{i=0}^t{s}_i{W}_{mk+l_i}}\right)}^{-1}$, respectively. In particular, for some special $a,b,p,q,m,t,s_i$ and $l_i\left(0\le i\le t\right)$, Theorem \ref{theorem 3.1} is just Theorems 2.1, 2.5-2.6 in \cite{A22} given by Yuan et al.