Question

Let \( \{x_n\}_{n \geq 1} \) be a sequence of positive real numbers. Which one of the following statements is always TRUE?

• A. If \( \{x_n\}_{n \geq 1} \) is a convergent sequence, then \( \{x_n\}_{n \geq 1} \) is monotone
• B. If \( \{x_n\}_{n \geq 1} \) is a convergent sequence, then the sequence \( \{x_n\}_{n \geq 1} \) does not converge
• C. If the sequence \( \{x_{n+1} - x_n\} \) converges to 0, then the series \( \sum_{m=1}^{\infty} x_m \) is convergent
• D. If \( \{x_n\}_{n \geq 1} \) is a convergent sequence, then \( e^{x_n} \) is also a convergent sequence

Hint:

For convergent sequences, remember that they must be both bounded and eventually monotone. If a sequence converges, it must become monotone after some point.

Answer 1

The Correct Option is D

Step 1: Understanding the sequence behavior.
For a sequence \( \{x_n\}_{n \geq 1} \) to converge, it must be bounded and monotonic. If a sequence converges, then its terms approach a limit and, by the properties of convergence, it becomes eventually monotone. Therefore, option (A) is true.
Step 2: Analyzing the options.
(A) If \( \{x_n\}_{n \geq 1} \) is a convergent sequence, then \( \{x_n\}_{n \geq 1} \) is monotone: Correct. A convergent sequence is always eventually monotone.
(B) If \( \{x_n\}_{n \geq 1} \) is a convergent sequence, then the sequence \( \{x_n\}_{n \geq 1} \) does not converge: Incorrect. This is contradictory, as the sequence is given to be convergent.
(C) If the sequence \( \{x_{n+1} - x_n\} \) converges to 0, then the series \( \sum_{m=1}^{\infty} x_m \) is convergent: Incorrect. Convergence of the difference sequence does not guarantee the convergence of the series.
(D) If \( \{x_n\}_{n \geq 1} \) is a convergent sequence, then \( e^{x_n} \) is also a convergent sequence: Incorrect. The sequence \( e^{x_n} \) may not converge if the terms of \( x_n \) grow without bound, even if \( x_n \) itself converges.
Step 3: Conclusion.
The correct answer is (A) If \( \{x_n\}_{n \geq 1} \) is a convergent sequence, then \( \{x_n\}_{n \geq 1} \) is monotone.