Infinite Number of Squares inside a Square
A series of squares are aligned along the diagonal of a unit square. Each square has edge length half the size of the previous square and is attached to the previous square at a corner. The sum of the areas of this series of squares is . As tends to infinity the sum converges to 1/3, as three such series fill the unit square. This is one way to see that the sum of the infinite series is 1/3.