Visualize Geometric Series

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 $$\frac{1}{4}+{\left(\frac{1}{4}\right)}^2+{\left(\frac{1}{4}\right)}^3+\cdots =\frac{3}{4}and\frac{4}{9}+{\left(\frac{4}{9}\right)}^2+{\left(\frac{4}{9}\right)}^3+\cdots =\frac{4}{5},$$ 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.