Problem 30 - Calculus: Early Transcendentals

Question

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{k = 1}^{\infty} \frac {k^2}{k^2 - 2k + 5} \)

Step-by-Step Solution

Answer

The series diverges because the terms do not approach zero as \( k \to \infty \).

Step 1: Analyze the General Term

The general term of our series is \( \frac{k^2}{k^2 - 2k + 5} \). We need to determine the behavior of this term as \( k \) approaches infinity. The highest degree term in both the numerator and the denominator is \( k^2 \), thus as \( k \to \infty \), \( \frac{k^2}{k^2 - 2k + 5} \approx \frac{k^2}{k^2} = 1 \). This indicates that the term does not approach zero.

Step 2: Apply the Divergence Test

The divergence test states that if \( \lim_{k \to \infty} a_k eq 0 \), or does not exist, then the series \( \sum_{k=1}^{\infty} a_k \) diverges. From Step 1, we found that \( \lim_{k \to \infty} \frac{k^2}{k^2 - 2k + 5} = 1 \). Since the limit is not zero, the series diverges.

Key Concepts

Divergence Test

When working with infinite series, the Divergence Test is one of the most fundamental tools to determine whether a series diverges. It's a quick initial check to see if further tests are necessary or not. If you're dealing with a series \( \sum_{k=1}^{\infty} a_k \), the Divergence Test requires you to find the limit of the general term, \( a_k \), as \( k \to \infty \).

In our example, the general term is \( \frac{k^2}{k^2 - 2k + 5} \). By examining how the terms behave as \( k \) approaches infinity, we calculated that the limit is 1, not zero. The Divergence Test states:

• If \( \lim_{k \to \infty} a_k eq 0 \) or the limit does not exist, then the series diverges.
• Conversely, if \( \lim_{k \to \infty} a_k = 0 \), it does not guarantee convergence. Further tests would be needed.

In this case, since the limit is 1, the series diverges. The Divergence Test confirms that no finite sum can be found for this series.

General Term Analysis

Analyzing the general term of a series is an essential step in deciding whether a series might converge or diverge. It often involves determining the dominant behavior of the terms as \( k \) becomes very large. Let's take our series from the exercise: \( \sum_{k=1}^{\infty} \frac{k^2}{k^2 - 2k + 5} \).

The general term here is \( \frac{k^2}{k^2 - 2k + 5} \). To find its limiting behavior, we look at the highest degree terms in the numerator and denominator. Both are \( k^2 \), which simplifies the expression for large \( k \) to \( \frac{k^2}{k^2} = 1 \). This tells us that the term behaves like 1 when \( k \) is very large. Understanding that the term approaches a constant instead of zero is critical, helping in the application of the Divergence Test.

General Term Analysis provides insights into the tendencies of series components, assisting greatly with subsequent convergence or divergence tests.

Limit of a Sequence

Understanding the limit of a sequence is crucial when dealing with series, as it helps with determining whether the series will converge or diverge. A sequence is basically a list of numbers, and finding its limit means identifying what number, if any, the sequence approaches as its index goes to infinity.

For the series \( \sum_{k=1}^{\infty} \frac{k^2}{k^2 - 2k + 5} \), the general term represents a sequence: \( a_k = \frac{k^2}{k^2 - 2k + 5} \). To see if the series converges, we examine the limit \( \lim_{k \to \infty} a_k \). As we calculated, as \( k \to \infty \), this limit is 1, not zero.

• If the limit of a sequence is zero, it doesn't guarantee convergence, but it is a necessary condition for convergence of a series.
• If the limit is anything other than zero, it confirms divergence of the series through the Divergence Test.

With this sequence approaching 1, not zero, we've shown the series must diverge. Understanding sequences and their limits supports valuable tests like the Divergence Test in series mathematics.