Symmetric Inequalities for Reciprocal Sums of Fibonacci Numbers

This paper establishes symmetric inequalities for the reciprocal sums of Fibonacci numbers. By using elementary methods, recurrence properties, Cassini’s identity, and the Fibonacci–Lucas connection, we deduce the precise bounds for these reciprocal sums and an asymptotic formula for the corresponding floor function. This work is novel in two key aspects: it is the first to combine reciprocals of linear recurrence sequences with the tail of the Riemann zeta function, and the first to derive symmetric inequalities for such reciprocal sums.