Problem 29 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the series converges or diverges. In this set of problems knowledge of the Limit Comparison Test is assumed. \(\sum_{k=3}^{\infty} \frac{k}{2 k^{3}-2}\)

Step-by-Step Solution

Answer

The given series converges.

Step 1: Simplify the given Series

First, simplify the series as much as possible. The given series is: \(\sum_{k=3}^{\infty} \frac{k}{2k^{3}-2}\)Simplify the denominator by factoring 2 out:\(\sum_{k=3}^{\infty} \frac{k}{2(k^{3}-1)}\)

Step 2: Apply Limit Comparison Test

Choose a simpler series that the given series can be compared to, typically if our series is \(a_k\), we can choose \(b_k = 1/k^p\). Here, choose \(b_k = 1/k^2\). The Limit Comparison Test states that if the limit of \(a_k/b_k\) as \(k→∞\) is a finite and positive number, both the series converge or both diverge.So it is time to calculate this limit:\(\lim_{k \to \infty} \frac{k / (2(k^{3}-1))}{1/k^2} = \lim_{k \to \infty} \frac{k^3}{2k^{3}-2}\)Appling l'Hopital's rule three times since the limit is \(0/0\), the limit becomes:\(\lim_{k \to \infty} \frac{6k}{6k^{2}} = 1\)Since limit is greater than zero and finite, so by Limit Comparison Test, the series behaves the same way as \(1/k^2\)

Step 3: Comparison Series Convergence

The series \(∑1/k^2\) is a p-series with p=2 which is >1, we know this is a convergent p-series. Therefore, by Limit Comparison Test, since the series we used for comparison converge, so does our series.

Key Concepts

Convergence of Series

Understanding the convergence of series is pivotal in calculus, as it helps us determine whether an infinite sum has a finite value. A series can either converge, which means its terms add up to a specific limit, or diverge, indicating that its sum is infinite or does not settle to any number.

When investigating convergence, we look at the behavior of the series as the number of terms increases indefinitely. We have various tests to determine convergence, such as the Limit Comparison Test illustrated in the exercise. In essence, this test compares our series to a simpler series whose behavior we already understand. If the two series can be shown to be similar in a certain mathematical way, we can conclude the convergence (or divergence) of the series in question based on the known behavior of the simpler series.

Remember, when using the Limit Comparison Test, the series we compare to should be chosen wisely. It should ideally simplify the analysis while still retaining the fundamental character of the original series.

p-series

A p-series is an infinite series of the form \(\sum_{n=1}^{\infty} \frac{1}{n^p}\), where \(p\) is a constant. The convergence of a p-series depends on the value of \(p\). For \(p > 1\), the series converges; and for \(p \leq 1\), it diverges.

Understanding p-series is important because they often serve as a comparison for other series, like in the textbook exercise we're discussing. When the chosen series for comparison is a p-series with \(p > 1\), this implies that we are dealing with a convergent comparison series, aiding us in deciding about the original series' behavior.

L'Hopital's Rule

L'Hopital's rule is a technique for evaluating limits that produce indeterminate forms, like \(\frac{0}{0}\) or \(\frac{\text{\infty}}{\text{\infty}}\). If the limit of a ratio of two functions results in one of these indeterminate forms, then l'Hopital's rule states that this limit can be found by taking the limit of the ratio of their derivatives, provided that the derivatives meet certain conditions.

Applying l'Hopital's rule as done in our exercise can often simplify complex expressions and aid in evaluating the limit more easily. This rule was used to find that the limit of the series, after simplification, was 1, leading to a conclusion about the series' convergence using the Limit Comparison Test.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. Understanding infinite series is key to many areas of mathematics and its applications. In our textbook problem, we deal with such a series and aim to decide on its convergence or divergence.

An infinite series can express a finite sum if it converges. Techniques and tests like the Limit Comparison Test help in analysing these series. The series converges if the sequence of partial sums approaches a specific value as the number of terms grows. Divergence, on the other hand, occurs when there is no such finite limit.

Seeing an infinite series might be daunting at first, but with the right tools and understanding of concepts such as p-series and l'Hopital's rule, we can dissect these series and reveal their nature.