The asymptotic behavior of the reciprocal sum of generalized Fibonacci numbers

Let ${\left(u_n\right)}_{n\ge 0}$ be the special Lucas $u$-sequence defined by

$$u_{n+2}=A{u}_{n+1}-B{u}_n,u_0=0,u_1=1,$$

where $n\ge 0$, $B=\pm 1$, and $A$ is an integer such that $A^2-4B>0$. Let

$$a_k=\frac{1}{u_{mk}^s},\frac{1}{u_{mk}+u_{mk+l}},\frac{1}{{\sum }_{i=0}^l{u}_{mk+i}},\frac{1}{u_{mk}{u}_{mk+2l}},\frac{1}{u_{mk}{u}_{mk+2l-1}},\frac{1}{u_{mk}+C},$$

where $m,l$ are positive integers, $s=1,2,3,4$, and $C$ is any constant. The aim of this paper is to find a form $g_n$ such that

$$\underset{n\to \mathrm{\infty }}{lim}\left({\left(\sum _{k=n}^{\mathrm{\infty }}{a}_k\right)}^{-1}-g_n\right)=0.$$

For example, we show that

$$\underset{n\to \mathrm{\infty }}{lim}\left({\left(\sum _{k=n}^{\mathrm{\infty }}\frac{1}{u_{mk}}\right)}^{-1}-\left(u_{mn}-u_{m(n-1)}\right)\right)=0.$$