Der Integraltest

The Integral Test establishes whether an infinite series converges or diverges by comparing its sum to the area under a related continuous curve defined by the function corresponding to the given series.

This method is essential when summing discrete terms directly is too difficult or impractical.

For example, a glow stick gradually dims over time, losing its light intensity exponentially. Starting at its brightest in the first hour, the glow stick gradually dims, creating an infinite list of hourly energy values that is difficult to sum directly.

On a graph, the energy emitted during each hour appears as a vertical bar—tall at first, then rapidly shorter. A smooth curve passes precisely through the top-right corner of each bar, modeling the energy emission rate as a continuous function.

The Integral Test applies here because this energy emission function is positive, continuous, and decreasing over time.

The area under the curve shows the total energy released by the glow stick. Based on the pattern of values, this area is defined by an improper integral that converges to a finite total, so the series also converges.