Convergence of a sinusoidal infinite series from Borwein, Bailey, and Girgensohn

TL;DR

This paper proves the convergence of a specific infinite series involving sinusoidal terms, using the irrationality measure of pi, addressing a question posed by Borwein, Bailey, and Girgensohn.

Contribution

The paper provides a rigorous proof of the convergence of a previously unresolved sinusoidal infinite series employing irrationality measures.

Findings

The series converges absolutely.

The proof utilizes the irrationality measure of pi.

This resolves an open question from Borwein, Bailey, and Girgensohn.

Abstract

Borwein, Bailey, and Girgensohn (2004) asked whether the following infinite series converges: the sum of $(\frac{2}{3} + \frac{1}{3} \sin n)^n / n$ over all positive integers $n$. We prove that their series converges. The proof uses the irrationality measure of $\pi$.