The Integral Test

The Integral Test is a method for determining whether an infinite series converges or diverges by comparing the series to an improper integral. It is especially useful when the terms of a series are difficult to add directly, but they follow the values of a related continuous function. Instead of summing infinitely many discrete terms one by one, the test studies the area under a curve that represents the same pattern of decrease.

A glow stick provides a useful example of this idea. After activation, it begins at a high level of brightness and gradually loses intensity over time. If the energy emitted during each hour is recorded, these hourly values form an infinite sequence. Adding all of these values would produce an infinite series representing the total energy released over time.

On a graph, each hourly energy value can be represented by a vertical bar. The bars are tallest at the beginning and become shorter as the glow stick dims. A smooth curve passing through the top-right corner of each bar models the energy emission rate as a continuous function. This continuous function allows the series to be analyzed using integration.

The Integral Test applies when the related function is positive, continuous, and decreasing on the interval being considered. These conditions match the glow stick model because the emitted energy remains positive, changes smoothly, and decreases over time.

The area under the curve represents the total continuous energy released by the glow stick. This area is described by an improper integral over an infinite interval. If the integral converges to a finite value, then the corresponding infinite series also converges. If the integral diverges, then the series diverges as well. In this case, because the glow stick’s energy output decreases rapidly and the area under the curve is finite, the infinite series of hourly energy values converges.