A Visual Proof of Lehmer’s Arctangent Sum

Abstract

SummaryWe provide a visual proof to Lehmer’s infinite sum of the arctangents of the inverse of the odd-indexed Fibonacci numbers. A few corollaries follow from the diagram, including Euler’s Machin-like formula and Strassnitzky’s formula.MSC: 11B39 Additional informationNotes on contributorsRex H. WuREX H. WU (MR Author ID: 1293646, ORCID 0000-0003-0970-3741) would like to thank the anonymous reviewer and the Editor for their many generous suggestions. Rex has been working with Garfield’s trapezoid for many years. He recently found more applications of it on the Fibonacci numbers. This article is one of them. Talking about Garfield’s trapezoid, which is named after president James A. Garfield for his proof on the Pythagorean theorem, Rex unexpectedly met his great-great-grandson Mr. Peter Garfield recently.