Problem 12 - Calculus: Early Transcendentals

Question

Determine whether the series converges or diverges. \( \displaystyle \sum_{k = 1}^{\infty} \frac {(2k - 1)(k^2 - 1)}{(k + 1)(k^2 + 4)^2} \)

Step-by-Step Solution

Answer

The series converges.

Step 1: Understand the Series

The series in question is \( \sum_{k = 1}^{\infty} \frac{(2k - 1)(k^2 - 1)}{(k + 1)(k^2 + 4)^2} \). We need to determine its convergence or divergence behavior.

Step 2: Simplify the Terms

Looking at the numerator \((2k - 1)(k^2 - 1)\), expand it: \[(2k - 1)(k^2 - 1) = 2k^3 - 2k - k^2 + 1\]This becomes \(2k^3 - k^2 - 2k + 1\). The denominator is \((k+1)(k^2+4)^2\), which is more complex but provides insight that a polynomial of higher degree in the denominator will affect convergence.

Step 3: Identify a Comparison Test

For large values of \(k\), the dominant terms in the expansion suggest the expression simplifies similarly to \( \frac{k^3}{k^5} \), as both the highest degree terms have a leading influence. This simplifies to \( \frac{1}{k^2} \). This is the form of a \( p\)-series with \( p = 2 \), which converges.

Step 4: Apply the Limit Comparison Test

Compare the series \( \frac{(2k - 1)(k^2 - 1)}{(k + 1)(k^2 + 4)^2} \) with \( \frac{1}{k^2} \). Apply the limit comparison test:\[ \lim_{k \to \infty} \frac{\frac{(2k - 1)(k^2 - 1)}{(k + 1)(k^2 + 4)^2}}{\frac{1}{k^2}} = \lim_{k \to \infty} \frac{(2k - 1)(k^2 - 1)k^2}{(k + 1)(k^2 + 4)^2} \]Upon simplifying the highest degree terms dominate, the denominator has higher power causing the limit to approximate a constant. If this constant is finite and nonzero, both series share convergence behavior.

Step 5: Conclude the Convergence

Given that the limit from Step 4 approaches a non-zero constant and \( \sum \frac{1}{k^2} \) is known to converge, our original series is convergent by the Limit Comparison Test.

Key Concepts

Comparison Test

The comparison test is a helpful tool in understanding whether a series converges or diverges. It relies on the idea that if one series behaves in a known way, another one sharing similar properties will behave similarly. The test comes in two main forms:

• If a series with non-negative terms is less than or equal to a known convergent series, it too will converge.
• If it is greater than or equal to a known divergent series, then it will diverge.

This method is intuitive because it leverages the behavior of easier-to-understand series to predict how more complex ones will act. This requires, however, a good choice of a "comparison" series.

Limit Comparison Test

When the direct comparison test is not easily applicable, the limit comparison test comes to the rescue. It compares an unknown series to a known series in a slightly different way using limits.

• Consider two series, say \( a_n \) and \( b_n \).
• The limit comparison test states that if \lim_{n \to \infty} \frac{a_n}{b_n} = c\, where \( c \) is a finite number and \( c > 0 \), then the series \( \sum a_n \) and \( \sum b_n \) either both converge or both diverge.

This is particularly useful because it allows us to skip the need for inequalities, only requiring a limit computation. It is a powerful way to judge convergence especially when dealing with more complicated series terms.

p-series

One of the most famous series in mathematics is the p-series. It looks like this: \( \sum_{k=1}^{\infty} \frac{1}{k^p} \).

• The key factor in determining the convergence is the value of \( p \).
• If \( p > 1 \), the series converges. If \( p \leq 1 \), it diverges.

The simplicity of checking if \( p \) is greater than 1 makes this a handy rule. In the exercise, by identifying that the series resembles a p-series with \( p = 2 \), the convergence of this series was confirmed. This immediate check greatly simplifies understanding complex series.

Polynomial Expansion

Polynomial expansion involves multiplying expressions with several terms, which leads to another polynomial. It can help identify dominant terms in a series.

• For example, expanding \( (2k - 1)(k^2 - 1) \) in the exercise gives \( 2k^3 - k^2 - 2k + 1 \).
• This focuses attention on the term with the highest degree, here \( 2k^3 \).

In convergence tests, these dominant terms influence the convergence behavior since they grow faster than others as \( k \) increases. By comparing these terms to a p-series, or using limit comparison with a similar known series, we gain insights on the convergence of our original series. Recognizing and applying polynomial expansion is thus a crucial step in series analysis.