Question

Let \( \{a_n\} \) and \( \{b_n\} \) be sequences of real numbers such that \( \{a_n\} \) is increasing and \( \{b_n\} \) is decreasing. Under which of the following conditions, the sequence \( \{a_n + b_n\} \) is always convergent?

• A. \( \{a_n\} \) and \( \{b_n\} \) are bounded sequences
• B. \( \{a_n\} \) is bounded above
• C. \( \{a_n\} \) is bounded above and \( \{b_n\} \) is bounded below
• D. \( a_n \to \infty \) and \( b_n \to -\infty \)

Hint:

For the sum of monotonic sequences to converge, both sequences must be bounded, with one sequence bounded above and the other bounded below.

Answer 1

The Correct Option is A, C

Step 1: Understanding the conditions.
The sequence \( \{a_n\} \) is increasing, and the sequence \( \{b_n\} \) is decreasing. For the sequence \( \{a_n + b_n\} \) to be convergent, both sequences must not grow too large in magnitude. Specifically, they must be bounded in such a way that their sum does not diverge.
Step 2: Analyzing the options.
- (A) \( \{a_n\} \) and \( \{b_n\} \) are bounded sequences: This condition is sufficient to guarantee convergence of the sum. If both sequences are bounded, the sum of the sequences will be bounded, and since they are monotonic, their sum will converge.
- (B) \( \{a_n\} \) is bounded above: This alone is not sufficient. \( \{b_n\} \) must also be bounded below for the sum to converge.
- (C) \( \{a_n\} \) is bounded above and \( \{b_n\} \) is bounded below: This condition ensures that the sum \( \{a_n + b_n\} \) is bounded, and since both sequences are monotonic, their sum will converge.
- (D) \( a_n \to \infty \) and \( b_n \to -\infty \): If \( a_n \to \infty \) and \( b_n \to -\infty \), the sum may not converge, so this is incorrect.
Step 3: Conclusion.
The correct answers are (A) and (C), as both conditions ensure the sum \( \{a_n + b_n\} \) converges.