Question

For what values of \(p\) does the integral \(\int_{2}^{\infty} \frac{d x}{x \ln ^{p} x}\) exist and what is its value (in terms of \(p\) )?

Short answer

\(\int_{2}^{\infty} \frac{d x}{x \ln^p x}\) Answer: The integral converges for \(p > 1\).

Step-by-step solution

Simplify the integral

Let's perform a substitution in order to simplify the integrand. Let \(u = \ln x\). Then, \(\frac{1}{x} = \frac{du}{dx}\). Our integral becomes: \(\int_{2}^{\infty} \frac{d x}{x \ln^p x} = \int_{\ln 2}^{\infty} \frac{d u}{u^p}\).

Apply the Integral Test for Convergence

Now, let's apply the integral test to check whether the integral converges or diverges. This involves determining the limit of the integral as it approaches infinity: \(\lim_{b \to \infty} \int_{\ln 2}^{b} \frac{d u}{u^p}\). If this limit is finite, then the integral converges; if the limit is infinite, then the integral diverges.

Determine the Boundaries for Convergence

To determine the boundaries for convergence, we need to find when the integral diverges and when it converges. If \(p=1\), the integral is the harmonic series, which diverges. So, we consider \(p \ne 1\). We can now integrate the integral: \(\int_{\ln 2}^{\infty} \frac{d u}{u^p} = \frac{1}{1-p}\left[u^{1-p}\right]_{\ln 2}^{\infty} = \lim_{b \to \infty} \frac{1}{1-p}\left[u^{1-p}\right]_{\ln 2}^{b} = \frac{1}{1-p}\left[\lim_{b \to \infty} (b^{1-p} - (\ln 2)^{1-p})\right]\). The limit converges if \(p > 1\) and diverges if \(p < 1\). Therefore, the integral converges for \(p > 1\) and diverges for \(p < 1\).

Evaluate the Integral

Now that we know the integral converges for \(p > 1\), let's find its value in terms of \(p\): \(\int_{\ln 2}^{\infty} \frac{d u}{u^p} = \frac{1}{1-p}\left[u^{1-p}\right]_{\ln 2}^{\infty} = \lim_{b \to \infty} \frac{1}{1-p}\left[u^{1-p}\right]_{\ln 2}^{b} = \frac{1}{1-p}\left[\lim_{b \to \infty} (b^{1-p} - (\ln 2)^{1-p})\right] = \frac{1}{p-1}(\ln 2)^{1-p}\). For \(p>1\), the integral converges to the value of \(\frac{1}{p-1}(\ln 2)^{1-p}\).

Key concepts

Convergence

Convergence is a fundamental concept in calculus and mathematical analysis. When we talk about the convergence of an integral like \( \int_2^\infty \frac{dx}{x \ln^p x} \), we're exploring whether this integral sums to a finite value or not. In simpler terms, convergence determines if the area under the curve, as the upper limit approaches infinity, remains finite.

• If the integral converges, it means the area is finite, and the sum can be calculated.
• If it diverges, the area is infinite, and the integral does not yield a meaningful value.

For our specific integral, we found that convergence depends on the value of \(p\). If \(p>1\), the integral converges, providing a finite result. However, if \(p \leq 1\), it diverges, indicating an infinite area. Understanding convergence helps in determining the boundaries and conditions of integrals in calculus.

Substitution Method

The substitution method is a powerful technique used to simplify complex integrals, making them easier to solve. In this exercise, the substitution \(u = \ln x\) transformed the integral into a more manageable form. By performing this substitution, our integral changed from \( \int_2^\infty \frac{dx}{x \ln^p x} \) to \( \int_{\ln 2}^{\infty} \frac{du}{u^p} \).

• This method helps convert a difficult problem into one that's easier to integrate.
• It involves choosing an appropriate substitution to streamline the process.

After finding \(u = \ln x\), the derivative \( \frac{du}{dx} = \frac{1}{x} \) helps rewrite the integral's differential \(dx\) in terms of \(du\). This simplification allows applying standard integration techniques to solve the problem effectively.

Integral Test

The Integral Test is a useful tool for determining the convergence or divergence of series and related integrals. By evaluating the improper integral associated with a series, we can infer the behavior of the series itself. In our exercise, the Integral Test was applied to \( \int_{\ln 2}^{b} \frac{du}{u^p} \), with \(b\) approaching infinity.

• The Integral Test links the behavior of a function and its corresponding series.
• If an improper integral converges, so does the series; similarly, if it diverges, the series does too.

This test was crucial in identifying that the integral converges for \(p>1\). It reveals how integrals evaluate the sum of areas under curves, paralleling the total sum of a corresponding series.

Harmonic Series

The harmonic series is a fascinating example in mathematics, often cited in discussions of divergence. It is defined as the sum \(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots\). In this context, it ties into our integral problem when \(p=1\). The series \( \int_2^\infty \frac{dx}{x \ln x} \) becomes harmonic, mimicking the behavior of the harmonic series.

• This series is crucial because it diverges, illustrating a case where an infinite sum does not settle to a finite number.
• The divergence of the harmonic series for \(p=1\) helps us conclude that the integral diverges in similar scenarios.

Understanding the harmonic series provides insight into why certain integrals or series do not converge, emphasizing the conditions needed for finding alternative solutions or adjustments in mathematical analysis.