Name: testing series with area under curves
Uploaded: 2026-06-08T09:38:57.969Z
Duration: 1 min 23 s
Channel: JoVE

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&#x2014;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.

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&#x2019;s energy output decreases rapidly and the area under the curve is finite, the infinite series of hourly energy values converges.

The Integral Test

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JoVE Core

Calculus

Unit

Sequences and Series

KEY TERMS AND CONCEPTS

testing series with area under curves

Testing series with area under curves helps determine whether an infinite series converges or diverges. The Integral Test compares a series to an improper integral. It is useful when the terms are hard to add directly but match a related continuous function.

The idea is to replace many discrete terms with the area under a curve. A glow stick gives a clear example. After it is activated, it starts bright and then slowly loses intensity over time. If the energy released each hour is recorded, those hourly values make an infinite sequence. Adding them gives an infinite series for the total energy released.

On a graph, each hourly value can be shown as a vertical bar. The bars are tallest at the start and get shorter as the glow stick dims. A smooth curve through the top-right corner of each bar models the energy emission rate as a continuous function. That function lets the series be studied with integration.

The Integral Test works when the related function is positive, continuous, and decreasing on the interval being used. The glow stick model fits these conditions because the energy stays 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 an improper integral over an infinite interval. If the integral converges to a finite value, the matching infinite series also converges. If the integral diverges, the series diverges too. In this example, the energy output drops quickly enough that the area is finite, so the infinite series of hourly energy values converges.

Testing Series with Area Under Curves

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Testing series with area under curves helps determine whether an infinite series converges or diverges. The Integral Test compares a series to an improper integral. It is useful when the terms are hard to add directly but match a related continuous function. The idea is to replace many discrete terms with the area under a curve. A glow stick gives a clear example. After it is activated, it starts bright and then slowly loses intensity over time. If the energy released each hour is recorded,...

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key terms and concepts

Learn how the Integral Test uses a related continuous function and improper integral to decide whether an infinite series converges or diverges.

Testing Series with Area Under Curves | JoVE K12

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calculus

Testing Series with Area Under Curves

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