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Magazine Chapter 8: Problem 47
Use the test of your choice to determine whether the following series converge. $$\sum_{k=1}^{\infty} \frac{k^{2}+2 k+1}{3 k^{2}+1}$$
Short Answer
Expert verified
Answer: Diverge
Step by step solution
01
Define the given series and comparison series
The given series is:
$$S = \sum_{k=1}^{\infty} \frac{k^{2}+2 k+1}{3 k^{2}+1}$$
The comparison series chosen is:
$$C = \sum_{k=1}^{\infty} \frac{k^2}{k^2}$$
02
Perform the Limit Comparison Test
To perform the Limit Comparison Test, we need to find the limit of \(\frac{S_{k}}{C_{k}}\) as \(k \to \infty\), where \(S_{k}\) and \(C_{k}\) are the k-th terms of series \(S\) and \(C\), respectively. If this limit exists and is positive, then both series will either converge or diverge together.
$$\lim_{k\to\infty}\frac{\frac{k^{2}+2 k+1}{3 k^{2}+1}}{\frac{k^2}{k^2}}$$
03
Simplify the expression
Now simplify the expression inside the limit:
$$\lim_{k\to\infty}\frac{k^{2}+2 k+1}{3 k^{2}+1} \cdot \frac{k^2}{k^2} = \lim_{k\to\infty}\frac{(k^{2}+2 k+1)k^2}{(3 k^{2}+1)k^2}$$
$$\lim_{k\to\infty}\frac{k^4+2k^3+k^2}{3k^4+k^2}$$
04
Evaluate the limit
Next, evaluate the limit as \(k \to \infty\):
$$\lim_{k\to\infty}\frac{k^4+2k^3+k^2}{3k^4+k^2} = \lim_{k\to\infty}\frac{1+\frac{2}{k}+\frac{1}{k^2}}{3+\frac{1}{k^2}}$$
Because \(\frac{2}{k}\) and \(\frac{1}{k^2}\) go to zero as \(k \to \infty\), the limit simplifies to:
$$\lim_{k\to\infty}\frac{1+\frac{2}{k}+\frac{1}{k^2}}{3+\frac{1}{k^2}} = \frac{1}{3}$$
05
Determine convergence
Since \(\lim_{k\to\infty}\frac{S_{k}}{C_{k}} = \frac{1}{3}\), and the limit is positive and finite, both series \(S\) and \(C\) either converge or diverge together.
Now, recall that the comparison series is:
$$C = \sum_{k=1}^{\infty} \frac{k^2}{k^2} =\sum_{k=1}^{\infty}1$$
This series is a harmonic series, which is known to diverge. Since the comparison series diverges, the original series \(S\) also diverges.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Convergence of Series
Series convergence is a fundamental concept in calculus, dealing with whether the sum of an infinite series approaches a finite limit. When we talk about series convergence, we're essentially interested in checking if adding an infinite number of terms results in a finite sum. This concept is crucial as it forms the backbone of many mathematical and real-world applications.
In mathematical notation, we express convergence as \( \sum_{k=1}^{\infty} a_k \), where \( a_k \) represents the terms in the series. If the sum of these terms approaches a finite value as \( k \to \infty \), the series is said to converge. Convergence often requires evaluating individual terms' behavior and using specific tests like the Limit Comparison Test, Ratio Test, or others.
To determine convergence, mathematicians often look at similar or simpler series whose behavior is known. By comparing the two series, one can infer about the convergence of the original series. This analysis provides valuable insights into the series' behavior without calculating potentially complex sums.
In mathematical notation, we express convergence as \( \sum_{k=1}^{\infty} a_k \), where \( a_k \) represents the terms in the series. If the sum of these terms approaches a finite value as \( k \to \infty \), the series is said to converge. Convergence often requires evaluating individual terms' behavior and using specific tests like the Limit Comparison Test, Ratio Test, or others.
To determine convergence, mathematicians often look at similar or simpler series whose behavior is known. By comparing the two series, one can infer about the convergence of the original series. This analysis provides valuable insights into the series' behavior without calculating potentially complex sums.
Divergence
Divergence is the opposite of convergence and is equally important in the study of series. When an infinite series diverges, this means that the sum of its terms does not approach a finite limit. Understanding divergence is essential because it helps us identify series that do not settle down to a particular value, preventing misinterpretations of mathematical results.
One common sign of divergence is when the series' terms do not approach zero. If \( a_k \) does not trend towards zero, the series \( \sum_{k=1}^{\infty} a_k \) is likely to diverge. This is backed by the Divergence Test, which states that if the limit of \( a_k \) as \( k \to \infty \) is not zero, the series definitely diverges.
The exercise given uses the Limit Comparison Test to conclude divergence. By comparing the original series with the harmonic series, it was found that both series diverge together. Key to this decision was the limit being positive and finite, reinforcing that these series share convergence or divergence outcomes. In this case, since the harmonic series diverges, the series \( S \) follows suit.
One common sign of divergence is when the series' terms do not approach zero. If \( a_k \) does not trend towards zero, the series \( \sum_{k=1}^{\infty} a_k \) is likely to diverge. This is backed by the Divergence Test, which states that if the limit of \( a_k \) as \( k \to \infty \) is not zero, the series definitely diverges.
The exercise given uses the Limit Comparison Test to conclude divergence. By comparing the original series with the harmonic series, it was found that both series diverge together. Key to this decision was the limit being positive and finite, reinforcing that these series share convergence or divergence outcomes. In this case, since the harmonic series diverges, the series \( S \) follows suit.
Harmonic Series
The harmonic series is one of the simplest yet fascinating infinite series known in mathematics. It is expressed as \( \sum_{k=1}^{\infty} \frac{1}{k} \) and is recognized for its tendency to diverge, despite its diminishing terms.
This series demonstrates that even when terms become smaller and smaller, the accumulation can still result in an infinite sum. The fact that the harmonic series diverges is crucial, especially in fields like analysis and number theory. It surfaces frequently in various mathematical contexts, reminding us that the behavior of series isn't always intuitive.
Understanding the harmonic series' divergence is vital for series analysis. It serves as a standard reference for comparing other series to determine their convergence or divergence. When using the Limit Comparison Test, as in our exercise, the harmonic series provides a benchmark. Since it diverges, comparing other series to it can help deduce their divergent nature, as seen in the given problem.
This series demonstrates that even when terms become smaller and smaller, the accumulation can still result in an infinite sum. The fact that the harmonic series diverges is crucial, especially in fields like analysis and number theory. It surfaces frequently in various mathematical contexts, reminding us that the behavior of series isn't always intuitive.
Understanding the harmonic series' divergence is vital for series analysis. It serves as a standard reference for comparing other series to determine their convergence or divergence. When using the Limit Comparison Test, as in our exercise, the harmonic series provides a benchmark. Since it diverges, comparing other series to it can help deduce their divergent nature, as seen in the given problem.
