Addendum to: On the rational approximation of the sum of the reciprocals of the Fermat numbers

TL;DR

This paper proves that for any integer base b ≥ 2, the sum of the reciprocals of Fermat numbers has an irrationality exponent exactly equal to 2, confirming a specific measure of irrationality.

Contribution

It establishes that the irrationality exponent of the sum of reciprocals of Fermat numbers in base b is exactly 2, extending previous results and confirming a precise measure of irrationality.

Findings

Irrationality exponent of the sum is exactly 2 for all integer bases b ≥ 2.

Confirms the measure of irrationality for the sum of reciprocals of Fermat numbers.

Builds on previous work to refine understanding of the sum's irrationality properties.

Abstract

As a corollary of the main result of our recent paper, {\em On the rational approximation of the sum of the reciprocals of the Fermat numbers} published in this same journal, we prove that for each integer $b\geq 2$ the irrationality exponent of $\sum_{n\geqslant 0} s_2(n)/b^n$ is equal to $2$.