The Integral Test

The Integral Test determines whether an infinite series converges or diverges by comparing it to an improper integral of a related continuous function. Use this method when summing discrete terms directly is impractical. The test applies when the function is positive, continuous, and decreasing over the interval being considered.

A glow stick emits energy that decreases over time, creating hourly values forming an infinite series. On a graph, vertical bars represent each hour's energy, with a smooth curve passing through their top-right corners. This continuous function models the energy emission rate, allowing the series to be analyzed using integration and the Integral Test.

The function must be positive, continuous, and decreasing on the interval being considered. These conditions ensure the area under the curve accurately represents the series behavior. When all three conditions are met, the convergence or divergence of the improper integral determines the convergence or divergence of the series.

The area under the curve represents the total continuous value accumulated by the function over an infinite interval. If this area, described by an improper integral, converges to a finite value, the corresponding infinite series also converges. If the integral diverges, the series diverges as well.

Many series have terms too complex to sum directly, making traditional summation impractical. The Integral Test converts this discrete problem into a continuous one by studying the area under a related curve. Integration techniques often provide clearer insight into convergence behavior than attempting to add infinitely many terms.

When an improper integral converges to a finite value, it means the area under the curve remains bounded despite extending over an infinite interval. This finite area indicates that the corresponding infinite series also converges to a finite sum. The glow stick example demonstrates this: rapid energy decrease produces finite total energy despite infinite time.

For a positive, continuous, decreasing function, if the improper integral converges, the series converges. If the integral diverges, the series diverges. This one-to-one correspondence allows you to determine series behavior by evaluating a single integral rather than analyzing the series directly.