Problem 21 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Explain why the hypothesis that \(f(x)\) is decreasing is important in the Integral Test.

Step-by-Step Solution

Answer

The hypothesis that \(f(x)\) is decreasing is important in the Integral Test to ensure the total area under the curve \(f(x)\) (from 1 to infinity) overestimates the sum of the series. This guarantees that the series and the integral used for comparison have similar quantities. If this condition was relaxed, the test would not provide accurate results.

Step 1: Understand the Integral Test

The Integral Test is a method used to ascertain if a series is convergent or divergent. It states that if a function \(f(x)\) is continuous, positive and decreasing for all \(x \geq 1\), and \(a_n = f(n)\), then the infinite series \(\sum_{n=1}^{\infty} a_n\) has the same nature (convergent or divergent) as the integral \(\int_{1}^{\infty} f(x)dx\).

Step 2: Analyse the Importance of a Decreasing Function

The function \(f(x)\) must be decreasing to ensure the areas of the rectangles formed under the curve overestimate the total area, when considering the integral \(\int_{1}^{\infty}f(x)dx\). If the function is increasing, the rectangles formed under the curve would underestimate the total area. This will violate the basic premise of the Integral Test: the nature (convergence/divergence) of the total area under the curve \(f(x)\) from 1 to infinity is the same as the nature of the series \(\sum_{n=1}^{\infty}a_n\). Therefore, the requirement that \(f(x)\) is decreasing is an essential component of the Integral Test.

Step 3: Conclusion

So, the condition that \(f(x)\) is decreasing is important in the Integral test to ensure we are comparing similar quantities: the sum of the terms of an infinite series and the integral of a function \(f(x)\) from 1 to infinity. If the function is not decreasing, then our comparison basis collapses, and the test fails to provide reliable or accurate results.

Key Concepts

Convergence

When we talk about convergence in the context of the Integral Test, we are referring to the behavior of an infinite series. An infinite series is said to be convergent if the sum of its infinite terms approaches a specific finite value. For students, it's easiest to think about this as the series settling down to a single value as you add more and more terms. The Integral Test helps determine the convergence of a series by relating it to an improper integral. If the integral of a function from 1 to infinity converges to a finite number, then the associated series converges as well. Understanding convergence is crucial because it tells us if the infinite sum we're dealing with is feasible. For practical applications, it decides if a lengthy computation or theoretical analysis is worth pursuing or will simply diverge infinitely without settling to a useful solution.

Decreasing Function

A function is considered decreasing if, for any two points, say \(x_1\) and \(x_2\), where \(x_1 < x_2\), the inequality \(f(x_1) \geq f(x_2)\) holds. This means that as you move along the x-axis to the right, the function values do not increase.For the Integral Test to work, having \(f(x)\) as a decreasing function is vital:

• It ensures that the series of rectangles' area which approximate the integral don't jolt upwards unexpectedly, which would misrepresent the total area under the curve.
• It keeps the relationship between the discrete sum of the series and the continuous integral intact.

The requirement for a decreasing function makes sure we have a fair and correct comparison, so the result of determining convergence or divergence through the Integral Test remains reliable.

Infinite Series

An infinite series is the sum of an infinite sequence of numbers. Typically denoted as \(\sum_{n=1}^{\infty} a_n\), it is built by adding up terms that continue indefinitely.Infinite series can either converge or diverge:

• If the series converges, the sum approaches a specific finite number, offering valuable insights and results.
• If it diverges, the sum does not settle down, meaning it either grows without bound or oscillates indefinitely.

The Integral Test utilizes the properties of an infinite series by comparing it with a definite integral. This helps us determine whether the infinite series we're examining has a finite sum or not. Recognizing the behavior of infinite series is essential in mathematics and its applications, including physics, engineering, and other sciences.