Problem 58 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that \(0

Step-by-Step Solution

Answer

For convergence, \(p < 1 < q\).

Step 1: Understand the Problem

You're given an integral from 0 to infinity, \( \int_{0}^{\infty} \frac{1}{x^{p}+x^{q}} d x \), which converges. You need to determine the relationship between \(p\) and \(q\) for this integral to converge. Here, \(p\) and \(q\) are positive numbers with \(0 < p < q\).

Step 2: Determine the Behavior at Zero and Infinity

Consider the behavior of the integrand \(\frac{1}{x^{p}+x^{q}}\) as \(x\to0\) and \(x\to\infty\). At \(x\to0\), the expression \(x^p\) dominates over \(x^q\), making the integrand approximately \(\frac{1}{x^p}\). As \(x\to\infty\), \(x^q\) dominates the integrand, leading it to behave like \(\frac{1}{x^q}\).

Step 3: Assess Convergence at Zero

Given the behavior near \(x=0\), where the integrand is approximately \(\frac{1}{x^p}\), this segment of the integral converges if \(\int_{0}^{a} x^{-p} dx\) converges. This integral converges when \(p < 1\).

Step 4: Assess Convergence at Infinity

Near \(x=\infty\), the integrand behaves like \(\frac{1}{x^q}\). The integral \(\int_{a}^{\infty} x^{-q} dx\) converges if \(q > 1\) because it behaves like a p-series with a power greater than 1. This implies for convergence at infinity, \(q > 1\) is necessary.

Step 5: Combine Results

For the original integral to converge, both conditions \(p < 1\) and \(q > 1\) must hold simultaneously. Therefore, \(p < 1 < q\).

Key Concepts

Convergence of Integrals

Understanding the convergence of integrals involves checking whether an integration process leads to a finite result. When we deal with an integral like \( \int_{0}^{\infty} \frac{1}{x^{p}+x^{q}} \, dx \), the task is to analyze this across its entire range to see if it produces a finite sum.

Convergence of an integral can be influenced by its integrand's behavior at the endpoints of the integration limit. In improper integrals, as in our example, we need to pay special attention to both extreme ends: as \(x\) tends to zero and as \(x\) tends to infinity.

For an integral to converge:

• The part of the integral evaluated at the lower limit does not approach infinity.
• Likewise, the part evaluated at the upper limit should also stay finite.

For the exercise given, the conditions at both endpoints, \( x \rightarrow 0 \) and \( x \rightarrow \infty \), play a pivotal role in ensuring the overall convergence.

Power Functions

Power functions are mathematical expressions of the form \( f(x) = x^n \). In our scenario, we examine \( x^p \) and \( x^q \), where \(p\) and \(q\) are exponents within the integrand.

The behavior of these power functions varies greatly depending on whether \(x\) is near zero or infinity:

• When \( x \) is close to zero, \( x^p \) grows quicker than \( x^q \) given \( 0 < p < q \). This is because smaller exponents result in larger values for small \( x \).
• Conversely, as \( x \) approaches infinity, \( x^q \) overtakes \( x^p \) in dominance due to larger exponents leading to more significant growth.

These characteristics are crucial for determining the convergence of the integral. The domination by \( x^p \) at \( x \approx 0 \) indicates we must ensure that \( p < 1 \) for convergence. At infinity, \( x^q \) offers guidance for convergence as long as \( q > 1 \). Thus, understanding how these power functions behave in different scenarios helps establish the integral's convergence conditions.

Calculus Problem-Solving

To tackle calculus problems like the one in our example, a structured approach is essential. Start by understanding the problem requirements—distilling it down to a matter of deciding when the integral converges.

In problem-solving with calculus, especially with improper integrals, follow these steps:

• Analyze the integrand: Examine the function being integrated to predict its behavior over the desired interval.
• Check end behavior: Assess how the function behaves near critical points such as zero and infinity. This helps in identifying critical conditions for convergence or divergence.
• Utilize known tests: Apply tests and known conditions (like p-series tests) to determine if the integral converges. In our case, ensuring \( p < 1 \) and \( q > 1 \) are necessary criteria.
• Proof and verify: Go through the logical proof to ensure the assumptions and tests applied accurately address the problem.

Approaching calculus problems methodically ensures that you don't miss critical factors, leading to a correct and reliable solution.