Problem 8 - Thomas Calculus

Question

Use the Integral Test to determine if the series in Exercises \(1-12\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=2}^{\infty} \frac{\ln \left(n^{2}\right)}{n} $$

Step-by-Step Solution

Answer

The series \( \sum_{n=2}^{\infty} \frac{\ln(n^2)}{n} \) diverges by the Integral Test.

Step 1: Identify the function and series

To apply the Integral Test, we start with the series \( \sum_{n=2}^{\infty} \frac{\ln(n^2)}{n} \). Notice that \( \ln(n^2) = 2\ln(n) \), so we can rewrite the series as \( \sum_{n=2}^{\infty} \frac{2\ln(n)}{n} \). The corresponding function is \( f(x) = \frac{2\ln(x)}{x} \).

Step 2: Verify conditions for the Integral Test

The Integral Test applies if \( f(x) \) is continuous, positive, and decreasing for \( x \geq 2 \). The function \( f(x) = \frac{2\ln(x)}{x} \) is continuous for \( x > 1 \) and is positive for all \( x \geq 2 \). We need to check if \( f(x) \) is decreasing by finding \( f'(x) \).

Step 3: Determine if the function is decreasing

Calculate the derivative \( f'(x) \) using the quotient rule: \[ f'(x) = \frac{ (2)(1/x)(x) - (2\ln(x))(1) }{x^2} = \frac{2 - 2\ln(x)}{x^2} \]. Simplifying gives \( f'(x) = \frac{2(1 - \ln(x))}{x^2} \). Since \( \ln(x) \geq 1 \) for \( x \geq e \), \( f(x) \) is decreasing for \( x \geq e \). For our series starting at \( n = 2 \), \( f(x) \) is decreasing as well.

Step 4: Set up the integral

Since all conditions are met, set up the integral \( \int_{2}^{\infty} \frac{2\ln(x)}{x} \, dx \) to determine the convergence of the series.

Step 5: Solve the integral

To solve \( \int \frac{2\ln(x)}{x} \, dx \), use integration by parts: let \( u = \ln(x) \), \( dv = \frac{2}{x} \, dx \). Then, \( du = \frac{1}{x} \, dx \) and \( v = 2 \ln(x) \). The integral becomes \( 2 \ln(x) \ln(x) - \int 2 \ln(x) \cdot \frac{1}{x} \, dx \), which further simplifies to evaluate as diverging to infinity.

Step 6: Conclusion based on the integral

Since the integral \( \int_{2}^{\infty} \frac{2\ln(x)}{x} \, dx \) diverges (as shown by evaluating that \( \lim_{b \to \infty} [ 2(\ln(x))^2/2 ]_{2}^{b} \) goes to infinity), the Integral Test indicates that the original series \( \sum_{n=2}^{\infty} \frac{\ln(n^2)}{n} \) diverges.

Key Concepts

Series Convergence

When evaluating series, one key question is whether it converges or diverges. Series convergence means that as you add up more terms, the total sum approaches a fixed value. In contrast, if a series diverges, the total sum becomes infinitely large or oscillates indefinitely.
In our exercise, we're asked to determine the convergence of the series \( \sum_{n=2}^{\infty} \frac{\ln(n^2)}{n} \).
To achieve this, we use the Integral Test. This test is handy because it allows us to determine the convergence of a series by comparing it to a related improper integral.

Continuous Function

For the Integral Test to be applicable, the function \( f(x) \) associated with the series must be continuous. A continuous function means it doesn’t have any breaks, jumps, or holes in its graph. This concept is crucial since any interruptions could affect the behavior of the integral needed for the test.
In this case, the function derived from the series is \( f(x) = \frac{2\ln(x)}{x} \). This function is continuous for \( x > 1 \) because both \( \ln(x) \) and \( \frac{1}{x} \) are continuous. Specifically, for our series starting at \( n = 2 \), the condition \( x \geq 2 \) conveniently ensures continuity.

Integration by Parts

Integration by parts is a technique to solve integrals that might not be straightforward. It follows from the product rule for derivatives and helps integrate products of functions. This method involves choosing parts of the function as \( u \) and \( dv \), then using the formula:

• \( \int u \, dv = uv - \int v \, du \)

Applying this to our problem with \( \int \frac{2\ln(x)}{x} \, dx \), we set \( u = \ln(x) \) and \( dv = \frac{2}{x} \, dx \). The integral becomes easier to handle since these choices simplify the expressions involved. Although the integration process is lengthy, understanding integration by parts is extremely helpful for many integral problems.

Derivative Calculation

When checking the conditions of the Integral Test, we need to confirm that the function \( f(x) = \frac{2\ln(x)}{x} \) is decreasing.
To do this, we calculate the derivative, using the quotient rule.
The quotient rule states:

• If \( f(x) = \frac{g(x)}{h(x)} \), then \( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{h(x)^2} \)

Substituting in our function, we find \( f'(x) = \frac{2(1 - \ln(x))}{x^2} \).
Further analysis shows \( f'(x) \) is negative for \( x \geq e \), confirming that the function is decreasing and meeting another criterion for the Integral Test's application.