The Irrationality of the Sum of Some Reciprocals

Abstract

Let {n _k } _k≥1 be an increasing sequence of positive integers. In this chapter we investigate some conditions under which the sum of the series

$$ \sum\limits_{k \geqslant 1} {\frac{1}{{{n_k}}}} $$

((10.1))

is an irrational number, and then we apply these results to the case for which the sequence {n _k } _k>1 is the sequence of Fermat numbers.

Satisfaction in science is not associated with solving a simple problem but comes from understanding complex situations and enjoying the small steps made to penetrate the mysteries of nature.

Victor Szebehely