Question

Let $(a_n)$ and $(b_n)$ be sequences of real number. Let $(\frac{a_n}{b_n})$ and $(a_n b_n)$ are convergent, then

• A. $(a_n)$ and $(b_n)$ both may be divergent
• B. $(a_n)$ and $(b_n)$ are always convergent
• C. $(a_n)$ and $(b_n)$ should converge
• D. $(a_n)$ and $(b_n)$ can not diverge

Hint:

To test general statements in analysis questions, try using the sequence \(\mathbf{(-1)^n}\) as a quick test case.
Its oscillating behavior makes it an excellent source of counterexamples for questions about boundedness and convergence.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We evaluate the convergence of combined sequence products and quotients using properties of limits.
While the product or quotient of two convergent sequences is always convergent, the product or quotient of two divergent sequences can sometimes converge.

Step 3: Detailed Explanation:
Let us test the statements by trying to construct a counterexample:
Can we find two divergent sequences \((a_n)\) and \((b_n)\) such that their quotient \(\left(\frac{a_n}{b_n}\right)\) and product \((a_n b_n)\) both converge?
1. Let us define two oscillating sequences:
\[ a_n = (-1)^n \] \[ b_n = (-1)^n \] 2. Both sequences are divergent because they oscillate infinitely between \(-1\) and \(1\):
\[ (a_n) = \{-1, 1, -1, 1, \dots\} \] \[ (b_n) = \{-1, 1, -1, 1, \dots\} \] 3. Now, let us calculate their quotient sequence \(\left(\frac{a_n}{b_n}\right)\):
\[ \frac{a_n}{b_n} = \frac{(-1)^n}{(-1)^n} = 1 \quad \text{for all } n \in \mathbb{N} \] This is a constant sequence \(\{1, 1, 1, \dots\}\), which converges to 1.
4. Next, let us calculate their product sequence \((a_n b_n)\):
\[ a_n b_n = (-1)^n \times (-1)^n = (-1)^{2n} = 1 \quad \text{for all } n \in \mathbb{N} \] This is also a constant sequence \(\{1, 1, 1, \dots\}\), which converges to 1.
This example proves that both the quotient and the product can converge even if both individual sequences \((a_n)\) and \((b_n)\) are divergent.
Therefore, \((a_n)\) and \((b_n)\) both may be divergent.

Step 4: Final Answer:
The correct option is that \((a_n)\) and \((b_n)\) both may be divergent, matching Option (A).