The geometric series

Here is a geometric interpretation of the geometric series

\[ 1 + r + r^2 + r^3 + \ldots = \frac{1}{1 - r} ~~~~ \mbox{ if } | r | < 1 \]

due to Bill Casselman.

The initial trapezoid is one unit wide, one unit high on the left and r units high on the right. When the next term is added in by clicking the Next button, a new trapezoid is constructed by scaling the last trapezoid built by r and placed next to the last trapezoid. The horizontal length of all the trapezoids is then the sum of the first terms of the series. This shows that the geometric series converges when |r| < 1 to $ \frac{1}{1-r} $ and diverges otherwise.