Question

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.

Accepted Answer

Step 1: Understand the given limits. We have two sequences $$a_n$$ and $$b_n$$ such that $$\lim_{n 	o \infty} a_n = 0$$ and $$\lim_{n 	o \infty} b_n = \infty. $$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 $$\lim_{n 	o \infty} a_n b_n = 0$$ necessarily holds. Since $$a_n$$ tends to zero and $$b_n$$ tends to infinity, the product is an indeterminate form of type $$0 	imes \infty. $$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 $$a_n b_n = \frac{1}{n} 	imes n = 1$$, 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 $$a_n b_n = \frac{1}{n^2} 	imes n = \frac{1}{n}$$, 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 $$\lim_{n 	o \infty} a_n = 0$$ and $$\lim_{n 	o \infty} b_n = \infty$$, then $$\lim_{n 	o \infty} a_n b_n = 0$$" is not necessarily true. It requires additional conditions to hold.

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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Key Concepts

Limits of Sequences

Indeterminate Forms

Counterexamples in Limit Analysis

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.