Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

(Comparison Test) If \(0 \leq f(x) \leq g(x)\) on \([a, \infty)\), it can be shown that the convergence of \(\int_{a}^{\infty} g(x) d x\) implies the convergence of \(\int_{a}^{\infty} f(x) d x\), and the divergence of \(\int_{a}^{\infty} f(x) d x\) implies the divergence of \(\int_{a}^{\infty} g(x) d x .\) Use this to show that \(\int_{1}^{\infty} \frac{1}{x^{4}\left(1+x^{4}\right)} d x\) converges. Hint: On \([1, \infty), 1 /\left[x^{4}\left(1+x^{4}\right)\right] \leq 1 / x^{4}\).

Step-by-Step Solution

Answer

\(\int_{1}^{\infty} \frac{1}{x^{4}(1+x^{4})} \, dx\) converges by the Comparison Test.

Step 1: Understand the problem

We need to determine whether the improper integral \( \int_{1}^{\infty} \frac{1}{x^{4}(1+x^{4})} \, dx \) converges using the comparison test. We will compare \( f(x) = \frac{1}{x^{4}(1+x^{4})} \) to \( g(x) = \frac{1}{x^{4}} \) and show that \( \int_{1}^{\infty} g(x) \, dx \) converges.

Step 2: Compare the functions

Note that for \( x \geq 1\), \( 1+x^{4} \geq 1 \), which implies that \( x^{4}(1+x^{4}) \geq x^{4}\). Thus, \( f(x) = \frac{1}{x^{4}(1+x^{4})} \leq \frac{1}{x^{4}} = g(x) \) for all \( x \geq 1 \).

Step 3: Check the convergence of the comparison function

Calculate the integral \( \int_{1}^{\infty} \frac{1}{x^{4}} \, dx \). This is a p-integral with \( p = 4 > 1\), which means it converges. To evaluate:\[\int_{1}^{\infty} \frac{1}{x^{4}} \, dx = \left[ -\frac{1}{3x^{3}} \right]_{1}^{\infty} = \lim_{b \to \infty} \left( -\frac{1}{3b^{3}} + \frac{1}{3} \right) = \frac{1}{3}.\]

Step 4: Apply the Comparison Test

Since \( f(x) \leq g(x) \) for all \( x \geq 1 \) and \( \int_{1}^{\infty} \frac{1}{x^{4}} \, dx \) converges, it follows by the Comparison Test that \( \int_{1}^{\infty} \frac{1}{x^{4}(1+x^{4})} \, dx \) also converges.

Key Concepts

Comparison Test

The Comparison Test is a powerful tool in calculus that helps us assess the behavior of improper integrals. Suppose you have two functions, \( f(x) \) and \( g(x) \), that you want to compare on a given interval like \([a, \infty)\). This test states that if \(0 \leq f(x) \leq g(x)\) for all \(x\) in our interval, then the following holds true:

• If \( \int_{a}^{\infty} g(x) \, dx \) converges, then \( \int_{a}^{\infty} f(x) \, dx \) also converges.
• If \( \int_{a}^{\infty} f(x) \, dx \) diverges, then \( \int_{a}^{\infty} g(x) \, dx \) does as well.

In practice, you want to choose a function \(g(x)\) that is simpler and whose behavior (convergence or divergence) you already understand. This simplification allows you to infer the behavior of a more complex function \(f(x)\) by comparison. This test is extremely useful when evaluating the convergence of more complex or unknown integrals.

Convergence and Divergence

When dealing with improper integrals, deciphering whether they converge or diverge is essential. Convergence of an integral means that it sums to a finite value as the upper limit approaches infinity. In contrast, divergence indicates the integral grows without bound or doesn't approach any finite value. Determining convergence or divergence is vital as it tells you about the behavior and limit of a function over an infinite interval.

In the exercise given, the aim was to determine whether the integral of \( \frac{1}{x^{4}(1+x^{4})} \) over \([1, \infty)\) converges. By using a known function for comparison, we can ascertain the behavior of the original function without directly evaluating the entire integral, making your task manageable and reducing complexity. Such evaluation provides insights into boundary behaviors and accumulation trends for various mathematical applications.

P-integrals

P-integrals, or the integrals of the form \( \int_{a}^{\infty} \frac{1}{x^p} \, dx \), serve as a critical reference point in determining convergence of functions, especially using the comparison test. For a p-integral:

• It converges if \( p > 1 \).
• It diverges if \( p \leq 1 \).

The rationale behind this can be understood through integration and limit analysis. In our specific problem, \( \int_{1}^{\infty} \frac{1}{x^4} \, dx \) was used to compare. Given that \( p = 4 > 1 \), the integral converges, making this function a useful comparison for evaluating the convergence of more complicated functions, such as \( \frac{1}{x^{4}(1+x^{4})} \).

Having a firm grasp of p-integrals is essential for calculus students, as they widely use these references in improper integral assessments, readily providing conclusions on numerical behavior.