On the sum of the reciprocals of k-generalized Fibonacci numbers

Abstract

Abstract In this note, we that if { F n ( k ) } n ≥ 0 {\left\{ {F_n^{\left( k \right)}} \right\}_{n \ge 0}} denotes the k-generalized Fibonacci sequence then for n ≥ 2 the closest integer to the reciprocal of ∑ m ≥ n 1 / F m ( k ) \sum\nolimits_{m \ge n} {1/F_m^{\left( k \right)}} is F n ( k ) − F n − 1 ( k ) F_n^{\left( k \right)} - F_{n - 1}^{\left( k \right)} .