Convergent Series of Rectangles to Fill a Unit Square

This Demonstration shows graphically how the sum of a certain infinite series converges to 1. Blue rectangles are successively added inside a square of area 1. Each iteration shades a fraction (numerator , denominator ) of the unshaded region of the square. The shaded rectangle has area given recursively by with , so that . The area after rectangles have been shaded is given recursively by with . Thus , which tends to 1 as .