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

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