  The aim of this paper is to give linear independence results for the values
of certain series. As an application, we derive arithmetical properties of the
sums of reciprocals of Fibonacci and Lucas numbers associated with certain
coprime sequences $\{n_\ell\}_{\ell\geq1}$. For example, the three numbers \[
1,\qquad\sum_{p\text{:prime}}^{}\frac{1}{F_{p^2}},\qquad\sum_{p\text{:prime}}^{}\frac{1}{L_{p^2}}
\] are linearly independent over $\mathbb{Q}(\sqrt{5})$, where $\{F_n\}$ and
$\{L_n\}$ are the Fibonacci and Lucas numbers, respectively.
