Problem 27 - Thomas Calculus

Question

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{\ln n}{n^{3}}$$

Step-by-Step Solution

Answer

The series converges by the Integral Test.

Step 1: Identify the Series

The given series is \( \sum_{n=1}^{\infty} \frac{\ln n}{n^3} \). We want to determine if this series converges or diverges.

Step 2: Apply the Integral Test

The Integral Test can be applied since \( f(n) = \frac{\ln n}{n^3} \) is positive, continuous, and decreasing for \( n \geq 2 \). Compute the improper integral \( \int_{2}^{\infty} \frac{\ln x}{x^3} \, dx \).

Step 3: Solve the Improper Integral

Integrate \( \int \frac{\ln x}{x^3} \, dx \) by parts. Let \( u = \ln x \) and \( dv = x^{-3} \, dx \). Then \( du = \frac{1}{x} \, dx \) and \( v = -\frac{1}{2}x^{-2} \). Apply the integration by parts formula \( \int u \, dv = uv - \int v \, du \).

Step 4: Evaluate the Integration by Parts

Substitute in the integration by parts formula: \( \int \frac{\ln x}{x^3} \, dx = -\frac{\ln x}{2x^2} + \int \frac{1}{2x^3} \, dx \). The second integral is \( -\frac{1}{4x^2} + C \). Evaluate from 2 to \( \infty \).

Step 5: Determine Convergence from the Integral Result

As \( x \to \infty \), both terms \( \frac{\ln x}{2x^2} \) and \( \frac{1}{4x^2} \) approach 0. Thus, \( \int_{2}^{\infty} \frac{\ln x}{x^3} \, dx \) converges. By the Integral Test, the series \( \sum_{n=1}^{\infty} \frac{\ln n}{n^3} \) also converges.

Key Concepts

Integral Test

The Integral Test is a powerful tool for determining the convergence of an infinite series. It applies only to series of non-negative terms, where the associated function is continuous, positive, and decreasing. In our case, we evaluate the series \( \sum_{n=1}^{\infty} \frac{\ln n}{n^3} \).To use the Integral Test:

• Identify the function \( f(n) = \frac{\ln n}{n^3} \) that corresponds to the terms of the series.
• Ensure the function is positive, continuous, and decreasing for \( n \geq 2 \).
• Evaluate the corresponding improper integral \( \int_{2}^{\infty} \frac{\ln x}{x^3} \, dx \).

If the improper integral converges, then the series converges. If the integral diverges, the series diverges too. This parallels the behavior of the sum and the integral beyond a certain point.

Improper Integral

Improper integrals extend the concept of definite integrals to unbounded regions or integrals with infinite limits. When applying the Integral Test, we solve the improper integral \( \int_{2}^{\infty} \frac{\ln x}{x^3} \, dx \), which needs special attention as it involves infinity.In this case:

• Identify the unbounded part of the integration, which starts from 2 and extends to infinity.
• Evaluate its limit to see if it converges or diverges.

Calculating this involves using techniques like integration by parts, simplifying the often complex expressions. Convergence of this integral hints at similar behavior for the associated series, making improper integrals essential for convergence tests.

Integration by Parts

Integration by Parts is an essential calculus technique to solve integrals that are products of functions. It rearranges the original integral into simpler parts that are easier to integrate. The formula involves two functions \(u\) and \(dv\):\[ \int u \, dv = uv - \int v \, du \]For \( \int \frac{\ln x}{x^3} \, dx \), pick:

• \( u = \ln x \)
• \( dv = x^{-3} \, dx \)

This gives us:

• \( du = \frac{1}{x} \, dx \)
• \( v = -\frac{1}{2}x^{-2} \)

Substitute these into the formula to break down the integral to a simpler form, simplifying towards a solution that determines convergence. This method fundamentally transforms difficult integrals into expressions easier to integrate or evaluate, playing a key role for solving many calculus problems.