On One Series of the Reciprocals of the Product of two Fibonacci Numbers Whose Indices Differ by an Even Number

Abstract

This paper is inspired by a very interesting YouTube video by Michael Penn, professor of mathematics at Randolph College in Virginia, USA. He dedicated himself to the popularization of mathematics on his website in addition to his teaching and scientific work at the university and in addition to his scientific work. First, we deal with four specific series of the reciprocals of the product of two Fibonacci numbers whose indices differ by 2, 4, 6, and 8. Then, we generalize these four results to the series of the reciprocals of the product of two Fibonacci numbers whose indices differ by an even number. Finally, we perform a numerical verification of the derived formula using Maple 2020 software. Based on the derived formula, it can be concluded that the series we are dealing with belong to infinite series whose sum can be expressed in closed form.