Problem 62 - Calculus Single Variable

Question

For which values of \(p\) is \(\int_{e}^{\infty} \frac{1}{x \ln ^{p}(x)} d x\) convergent?

Step-by-Step Solution

Answer

The integral converges when \( p > 1 \).

Step 1: Understand Convergence Criterion for Improper Integrals

The integral is of the form \( \int_a^\infty f(x) \, dx \). For this integral to converge, \( \lim_{{b \to \infty}} \int_a^b f(x) \, dx \) must exist and be finite. We'll apply this criterion to \( f(x) = \frac{1}{x \ln^p(x)} \) and analyze the behavior as \( x \to \infty \).

Step 2: Set Up the Integral

We need to evaluate \( \int_e^b \frac{1}{x \ln^p(x)} \, dx \) as \( b \to \infty \). Before taking the limit, consider substituting \( u = \ln(x) \) which simplifies the integral. Note that \( du = \frac{1}{x} dx \), so \( dx = x \cdot du \).

Step 3: Substitute Variables

With \( u = \ln(x) \), \( x = e^u \), and \( dx = e^u \, du \). Express the integral in terms of \( u \): \[ \int_{1}^{\ln(b)} \frac{1}{e^u u^p} e^u \, du = \int_{1}^{\ln(b)} \frac{1}{u^p} \, du. \] This transforms the problem into determining the convergence of the integral \( \int_{1}^{\ln(b)} \frac{1}{u^p} \, du \).

Step 4: Analyze the Integral with respect to p

The integral \( \int_{1}^{\ln(b)} \frac{1}{u^p} \, du \) resembles the integral for a p-series. This integral will converge if and only if \( p > 1 \). Therefore, determine convergence based on the value of \( p \).

Step 5: Establish Condition for Convergence

If \( p > 1 \), then the integral \( \int_1^{\ln(b)} \frac{1}{u^p} \, du \) converges and hence the original integral converges. If \( p \leq 1 \), the integral will diverge. Therefore, the convergence of the given integral depends on \( p > 1 \).

Key Concepts

Convergence Criteria

When dealing with improper integrals, understanding convergence criteria is crucial. Usually, these integrals extend to infinity, making it important to determine if the integral's value culminates to a specific finite number, or keeps growing endlessly.

For the integral \(\int_a^\infty f(x) \ dx\), the process involves checking if \(\lim_{{b \to \infty}} \int_a^b f(x) \ dx\) results in a finite limit. If it does, the integral is convergent. Otherwise, it is divergent.

In our specific exercise, we observe this criterion applied to \(f(x) = \frac{1}{x \ln^p(x)}\). The quest is to conclude the integral \(\int_{e}^{\infty} \frac{1}{x \ln ^{p}(x)} \ dx\) based on its behavior as \(x\) approaches infinity. Yet, before simplifying, the strategy shifts to modifying the integral integration range.

Substitution Method

The substitution method simplifies complex integration problems, particularly when dealing with improper integrals. The process involves replacing a difficult variable with an easier one, transforming the integral into a more manageable form.

In this example, we define the substitution with \(u = \ln(x)\), leading to \(x = e^u\)and \(dx = e^u \, du\).

By transforming the variable like this, the equation shifts to:

• Convert \(\int_{e}^{b} \frac{1}{x \ln^p(x)} dx\) to \(\int_{1}^{\ln(b)} \frac{1}{u^p} \, du\).

This simplified version highlights any potential for convergence, reducing the complexity inherent in the original integral.

P-Series Test

A powerful tool for understanding convergence is the p-series test. The p-series is characterized by the integral: \(\int_{a}^{\infty} \frac{1}{u^p} \, du\). This series converges only under specific circumstances: when\(p > 1\).

Convergence analysis of the p-series reveals:

• If \(p > 1\), the series converges.
• If \(p \leq 1\), the series diverges.

In this particular task, we reshaped the problem into one fitting the p-series model.
By converting to \(\int_{1}^{\ln(b)} \frac{1}{u^p} \, du\), we assess convergence based solely on the parameter \(p\). If \(p > 1\), the original integral converges due to the p-series test.

Limits and Infinity

Understanding limits and infinity is key when tackling improper integrals. These concepts enable mathematicians to evaluate behavior as variable approaches vast numbers.

Particularly in integrals extending to infinity, limits provide insights into whether a particular function gathers around a specific value (converges) or not (diverges). As \(x\) increases without bound, \(f(x)\) behavior is scrutinized to determine convergence.

Applying this knowledge to our problem, \(\lim_{{b \to \infty}} \int_{e}^{b} \frac{1}{x \ln^p(x)} dx\), we examine if a limit exists.
Eventually, whether \(p > 1\) proves decisive, marking the point of gradual convergence, as further confirmed by the p-series test.