Visualize Geometric Series

TL;DR

This paper reviews and analyzes visual proofs of the geometric series formula from multiple sources, highlighting their uniqueness and common ideas, thereby deepening understanding of geometric series visualization.

Contribution

It identifies the unique features of visual proofs for geometric series and connects different proof methods through shared ideas, enhancing pedagogical approaches.

Findings

Proofs are unique under certain criteria.

Common ideas link different visual proof methods.

Visual proofs effectively illustrate geometric series convergence.

Abstract

We review Mabry's, Edgar's, and the Viewpoints 2000 Group's proofs without words for the geometric series formula. Mabry and Edgar proved without words that $$\frac{1}{4} + \left(\frac{1}{4}\right)^2 + \left(\frac{1}{4}\right)^3 + \cdots\ =\ \frac{3}{4}\quad\mbox{ and }\quad\frac{4}{9} + \left(\frac{4}{9}\right)^2 + \left(\frac{4}{9}\right)^3 + \cdots\ =\ \frac{4}{5},$$ respectively. We show that their proofs satisfy certain requirements that make them unique. We then illustrate a common idea between their and the Viewpoints 2000 Group's proofs.