Problem 8 - Essential Calculus

Question

Use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=2}^{\infty} \frac{\ln n}{n^{3}} $$

Step-by-Step Solution

Answer

The series \( \sum_{n=2}^{\infty} \frac{\ln n}{n^{3}} \) is convergent.

Step 1: Rewrite the series as a function

The series can be rewritten as a function \( f(x) = \frac{\ln x}{x^{3}} \).

Step 2: Apply the Integral Test

To apply the Integral Test, integrate \( f(x) \) from 2 to infinity:\n \[ \int_{2}^{\infty} \frac{\ln x}{x^{3}} dx \].

Step 3: Evaluate the integral

To evaluate this integral, we can use integration by parts. Let \( u = \ln x, dv = \frac{dx}{x^{3}} \). Then \( du = \frac{dx}{x}, v = -\frac{1}{2x^{2}} \).\n After substituting these into the integral, we have:\n \[ -\frac{1}{2} \Bigg[\frac{\ln x}{x^{2}}\Bigg]_{2}^{\infty} + \frac{1}{2} \int_{2}^{\infty} \frac{1}{x^{2}} dx \] \n Evaluate both terms separately. The first term tends towards 0 as \( x \) approaches infinity and is finite for \( x = 2 \). The second term is a proper integral and its value is finite. Therefore, the original integral is finite.

Step 4: Determine the convergence

Since the integral of \( f(x) \) from 2 to infinity is finite, the Integral Test tells us that the original series \( \sum_{n=2}^{\infty} \frac{\ln n}{n^{3}} \) is convergent.

Key Concepts

Convergence and Divergence of Series

Understanding whether a series converges or diverges is crucial in calculus. A convergent series adds up to a finite sum, while a divergent series does not. To determine convergence or divergence, we can use various tests, including the Integral Test.

• Convergence implies that as you sum more and more terms of the series, you approach a certain limit.
• Divergence means that the sum of the series grows without bound or oscillates, never settling at any particular value.

The Integral Test connects series and integrals by evaluating if the integral of the function (related to series terms) from a point to infinity is finite. If finite, the series converges; if not, it diverges.

Integration by Parts

Integration by parts is a helpful technique for solving integrals involving products of functions. It is a calculus equivalent of the product rule for differentiation.

• To use it, choose functions \( u \) and \( dv \) such that the integral becomes easier after applying the formula \( \int u \, dv = uv - \int v \, du \).
• Selecting \( u = \ln x \) and \( dv = \frac{dx}{x^3} \) makes the integral manageable because \( du = \frac{dx}{x} \) while \( v = -\frac{1}{2x^2} \).

This method helps in transforming a complex integral into a form that is simpler to evaluate. Using the right combination of \( u \) and \( dv \) is key to simplifying the problem.

Improper Integrals

Improper integrals are integrals with one or more infinite limits or involving unbounded integrands. They require a careful approach, usually by taking limits.

• The integral used in the Integral Test is improper because it starts at 2 and goes to infinity, giving it an infinite limit.
• Evaluating an improper integral involves checking if this integral approaches a finite value, ensuring convergence.

In the example, transforming the original integral into parts and showing that each part results in a finite value demonstrates its convergence. Improper integrals extend the concept of integration beyond finite intervals.

Series in Calculus

Series are fundamental in calculus, representing the sum of sequences of terms. They appear in various forms, like arithmetic or geometric series, and have significant applications.

• In calculus, series allow us to approximate functions, evaluate constants, and model various phenomena.
• Understanding the behavior of series concerning convergence or divergence underpins much of mathematical analysis.

The series \( \sum_{n=2}^{\infty} \frac{\ln n}{n^3} \) in this exercise is a specific case of how series might behave. Applying the Integral Test gives us insights into its behavior, affirming that math often mirrors nature with its precision and predictability.