Convergence And Divergence Analysis If A_n And B_n Are Greater Than 0

In the fascinating realm of mathematical analysis, sequences and series hold a central position. Understanding their behavior, particularly their convergence or divergence, is crucial for a wide array of applications. This article delves into a specific scenario involving two sequences, a_n and b_n, both of which are strictly positive for all n greater than or equal to 1. We are given that the limit as n approaches infinity of the ratio b_n to a_n is 0. Our goal is to explore the implications of this condition on the convergence or divergence of the series formed by these sequences, namely ∑a_n and ∑b_n. The relationship between sequences and series is fundamental to many areas of mathematics, including calculus, real analysis, and complex analysis. Convergence and divergence are key concepts in these fields, determining whether an infinite sum has a finite value or grows without bound. The given condition, ${\lim_{n \to \infty} \frac{b_n}{a_n} = 0}$, suggests that b_n becomes significantly smaller than a_n as n increases. This intuition provides a starting point for analyzing the behavior of the series. We will examine how this relative size difference impacts the convergence or divergence of the respective series. Understanding these relationships is not only theoretically important but also has practical applications in various fields such as physics, engineering, and computer science, where infinite sums often arise in modeling complex systems. The initial statement presents a comparative relationship between two series based on the asymptotic behavior of their terms. Specifically, it explores the scenario where the terms of one series (b_n) become infinitesimally smaller relative to the terms of another series (a_n). This article aims to rigorously investigate the implications of this relationship on the convergence or divergence of the series. This involves examining relevant convergence tests and theorems, and applying them to the given conditions. By carefully dissecting the problem and leveraging established mathematical principles, we can draw definitive conclusions about the behavior of these series. The analysis will not only provide a solution to the specific problem at hand but also offer insights into the broader principles governing the convergence and divergence of series.

Problem Statement

If a_n > 0 and b_n > 0 for all n ≥ 1, and ${\lim_{n \to \infty} \frac{b_n}{a_n} = 0}$, which of the following statements is true?

a. ∑a_n and ∑b_n both converge or both diverge. b. If ∑a_n converges, then ∑b_n converges. c. If ∑b_n converges, then ∑a_n converges. d. If ∑a_n diverges, then ∑b_n diverges.

This question delves into the heart of series convergence and divergence, exploring how the asymptotic relationship between two sequences affects the behavior of their corresponding series. The condition ${\lim_{n \to \infty} \frac{b_n}{a_n} = 0}$ is particularly insightful, as it implies that the terms b_n become negligible compared to a_n as n approaches infinity. Understanding this relative behavior is key to determining the correct statement. The options presented cover various scenarios, linking the convergence or divergence of one series to that of the other. Option (a) suggests a strong correlation, stating that both series either converge or diverge together. This is a significant claim that requires careful examination. Options (b) and (c) propose conditional relationships, where the convergence of one series implies the convergence of the other. These options highlight the importance of the direction of implication and whether the condition ${\lim_{n \to \infty} \frac{b_n}{a_n} = 0}$ imposes a specific constraint on this relationship. Option (d) presents a scenario where the divergence of ∑a_n leads to the divergence of ∑b_n. This option challenges our understanding of how the divergence of a series with larger terms (a_n) affects the series with comparatively smaller terms (b_n). To solve this problem effectively, we need to draw upon fundamental convergence tests and theorems, such as the Comparison Test, the Limit Comparison Test, and the properties of convergent and divergent series. We will need to carefully consider the implications of the given condition on these tests and how they apply to the specific context of this problem. Furthermore, constructing counterexamples can be a powerful technique for disproving certain statements. By finding specific sequences a_n and b_n that satisfy the given condition but contradict one of the options, we can eliminate that option as a possibility. The problem encourages a deep understanding of the interplay between sequences and series, and how the asymptotic behavior of sequence terms dictates the convergence or divergence of the corresponding series. It is not merely about memorizing convergence tests but about applying them judiciously and understanding their underlying principles.

Solution and Explanation

The correct answer is b. If ∑a_n converges, then ∑b_n converges.

Explanation:

Given that ${\lim_{n \to \infty} \frac{b_n}{a_n} = 0}$, this means that for any ε > 0, there exists an integer N such that for all n > N, we have:

${|\frac{b_n}{a_n} - 0| < \epsilon}$

Which simplifies to:

${\frac{b_n}{a_n} < \epsilon}$

Or:

${b_n < \epsilon a_n}$

This inequality is the cornerstone of our argument. It states that beyond a certain point (N), each term b_n is smaller than ε times the corresponding term a_n, where ε can be chosen arbitrarily small. This relationship is crucial for applying the Comparison Test. The Comparison Test is a powerful tool for determining the convergence or divergence of a series by comparing it to another series whose behavior is known. In this case, we are given that ∑a_n converges. Our goal is to use this information, along with the inequality b_n < ε a_n, to deduce the convergence of ∑b_n. Since ∑a_n converges, we know that the sum of its terms approaches a finite limit. Multiplying each term by a positive constant ε does not change this fundamental property; the series ∑εa_n also converges. Now, we have two series, ∑b_n and ∑εa_n, and we know that b_n < ε a_n for all n > N. This satisfies the conditions of the Comparison Test, which states that if a series is term-by-term less than a convergent series, then it must also converge. Therefore, we can confidently conclude that ∑b_n converges. This result highlights the power of the Comparison Test in analyzing the convergence of series. It allows us to leverage the known behavior of one series to infer the behavior of another, provided we can establish a suitable term-by-term comparison. In this case, the condition ${\lim_{n \to \infty} \frac{b_n}{a_n} = 0}$ provided the critical link, enabling us to bound the terms of ∑b_n by the terms of a convergent series, thereby proving its convergence. The beauty of this proof lies in its simplicity and elegance. It leverages a fundamental definition of limits and a well-established convergence test to arrive at a definitive conclusion. This approach not only solves the specific problem but also reinforces the importance of understanding the underlying principles of convergence and divergence in mathematical analysis.

Let's break down why the other options are incorrect:

• a. ∑a_n and ∑b_n both converge or both diverge. This is not necessarily true. We have shown that if ∑a_n converges, then ∑b_n converges. However, if ∑a_n diverges, ∑b_n could still converge. Consider a_n = 1/n and b_n = 1/n². Here, ∑a_n diverges (harmonic series), but ∑b_n converges (p-series with p = 2).
• c. If ∑b_n converges, then ∑a_n converges. This is also not necessarily true. Using the same counterexample as above, ∑b_n converges, but ∑a_n diverges.
• d. If ∑a_n diverges, then ∑b_n diverges. This is false, as demonstrated by the counterexample in the explanation for option (a).

These counterexamples are crucial in demonstrating the limitations of the initial condition. While ${\lim_{n \to \infty} \frac{b_n}{a_n} = 0}$ does provide valuable information about the relative sizes of the terms, it does not guarantee a bidirectional relationship between the convergence or divergence of the series. The convergence of ∑a_n forces the convergence of ∑b_n, but the converse is not necessarily true. Similarly, the divergence of ∑a_n does not preclude the convergence of ∑b_n. These nuances are essential to grasp for a thorough understanding of series convergence. Constructing counterexamples is a powerful technique in mathematical analysis for disproving general statements. By finding specific cases that violate a proposed rule, we can demonstrate that the rule is not universally applicable. In the context of series convergence, counterexamples can help us refine our understanding of the conditions under which certain convergence tests and theorems hold true. The examples provided here, using the harmonic series and the p-series, are classic illustrations of how series with seemingly similar terms can exhibit drastically different convergence behavior. They highlight the importance of considering the rate at which the terms approach zero and how this rate affects the overall sum of the series. The analysis of these counterexamples further reinforces the understanding that the condition ${\lim_{n \to \infty} \frac{b_n}{a_n} = 0}$ establishes a one-way implication: the convergence of ∑a_n implies the convergence of ∑b_n. However, it does not provide sufficient information to infer the behavior of ∑a_n based on the behavior of ∑b_n. This nuanced understanding is crucial for applying convergence tests correctly and avoiding common pitfalls in series analysis.

Conclusion

In conclusion, given that a_n > 0, b_n > 0 for all n ≥ 1, and ${\lim_{n \to \infty} \frac{b_n}{a_n} = 0}$, the only true statement is that if ∑a_n converges, then ∑b_n converges. This is a direct consequence of the Comparison Test, which allows us to compare the terms of ∑b_n to a convergent series ∑εa_n. The other options are incorrect, as demonstrated by counterexamples involving the harmonic series and p-series.

This exploration underscores the importance of understanding convergence tests and the subtle relationships between sequences and series. The condition ${\lim_{n \to \infty} \frac{b_n}{a_n} = 0}$ provides a valuable piece of information about the relative sizes of the terms, but it does not guarantee a symmetrical relationship between the convergence or divergence of the series. The Comparison Test, in this case, allows us to leverage this information effectively, but it is crucial to remember that the converse implications do not necessarily hold true. The use of counterexamples further reinforces this understanding, demonstrating the limitations of the initial condition and highlighting the importance of rigorous proof in mathematical analysis. The interplay between sequences and series is a central theme in calculus and real analysis, and a deep understanding of these concepts is essential for tackling more advanced topics. The problem discussed here serves as a valuable illustration of the nuances involved in determining convergence and divergence and the power of fundamental theorems like the Comparison Test. It also emphasizes the importance of critical thinking and the ability to construct counterexamples to challenge assumptions and refine our understanding. The process of analyzing this problem not only provides a solution but also reinforces the broader principles governing the behavior of infinite sums and the conditions under which we can make definitive conclusions about their convergence or divergence. The exploration of these concepts is not merely an academic exercise but has practical implications in various fields where infinite sums arise, such as physics, engineering, and computer science. A solid foundation in series convergence is therefore a valuable asset for anyone pursuing these disciplines.