Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.

b.If limₙ→∞aₙ = 0 and limₙ→∞bₙ = ∞, then limₙ→∞aₙbₙ = 0.

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Step 1: Understand the given limits. We have two sequences \(a_n\) and \(b_n\) such that \(\lim_{n \to \infty} a_n = 0\) and \(\lim_{n \to \infty} b_n = \infty\). This means \(a_n\) approaches zero and \(b_n\) grows without bound as \(n\) becomes very large.

Step 2: Analyze the product \(a_n b_n\). The question asks if \(\lim_{n \to \infty} a_n b_n = 0\) necessarily holds. Since \(a_n\) tends to zero and \(b_n\) tends to infinity, the product is an indeterminate form of type \(0 \times \infty\). This means the limit could be zero, infinite, or some finite number depending on the rates at which \(a_n\) approaches zero and \(b_n\) grows.

Step 3: Consider a counterexample to test the statement. For instance, if \(a_n = \frac{1}{n}\) and \(b_n = n\), then \(a_n b_n = \frac{1}{n} \times n = 1\), which does not tend to zero but to 1. This shows the statement is not always true.

Step 4: Alternatively, if \(a_n = \frac{1}{n^2}\) and \(b_n = n\), then \(a_n b_n = \frac{1}{n^2} \times n = \frac{1}{n}\), which tends to zero. This shows the limit can be zero in some cases.

Step 5: Conclusion: Since the product limit depends on the relative rates of \(a_n\) and \(b_n\), the statement "If \(\lim_{n \to \infty} a_n = 0\) and \(\lim_{n \to \infty} b_n = \infty\), then \(\lim_{n \to \infty} a_n b_n = 0\)" is not necessarily true. It requires additional conditions to hold.

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주요 개념

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Limits of Sequences

The limit of a sequence describes the value that the terms of the sequence approach as the index goes to infinity. Understanding how to evaluate limits of sequences is fundamental to analyzing their long-term behavior and determining convergence or divergence.

Indeterminate Forms

An indeterminate form arises when the limit of a product or quotient involves conflicting behaviors, such as one factor approaching zero and another approaching infinity. In such cases, the limit cannot be directly concluded and requires further analysis or counterexamples.

Counterexamples in Limit Analysis

Counterexamples are specific cases that disprove a general statement. When evaluating limit statements, constructing or identifying sequences that violate the proposed limit helps demonstrate whether the statement is true or false.

Explain why or why notDetermine whether the following statements are true and give an explanation or counterexample.b.If limₙ→∞aₙ = 0 and limₙ→∞bₙ = ∞, then limₙ→∞aₙbₙ = 0.