Question

Let \( \{x_n\}_{n \geq 1} \) be a sequence defined as \[ x_n = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}} - 2(\sqrt{n} - 1). \] Then which of the following options is/are correct?

• A. The sequence \( \{x_n\}_{n \geq 1} \) is unbounded.
• B. The sequence \( \{x_n\}_{n \geq 1} \) is monotonically decreasing.
• C. The sequence \( \{x_n\}_{n \geq 1} \) is bounded but does not converge.
• D. The sequence \( \{x_n\}_{n \geq 1} \) converges.

Hint:

When analyzing sequences involving sums, compare the growth rates of the terms involved and use asymptotic approximations to predict the sequence's behavior as \( n \to \infty \).

Answer 1

The Correct Option is B, D

Step 1: Analyze the sequence. The sequence is given as: \[ x_n = \sum_{k=1}^{n} \frac{1}{\sqrt{k}} - 2(\sqrt{n} - 1). \] The term \( 2(\sqrt{n} - 1) \) is a term that grows like \( 2\sqrt{n} \) as \( n \) increases, and the sum \( \sum_{k=1}^{n} \frac{1}{\sqrt{k}} \) grows like \( 2\sqrt{n} \) for large \( n \). 
Step 2: Behavior of the sequence for large \( n \). We approximate the sequence for large \( n \): \[ x_n \sim \sum_{k=1}^{n} \frac{1}{\sqrt{k}} - 2\sqrt{n}. \] Using the asymptotic approximation for the sum, we have: \[ \sum_{k=1}^{n} \frac{1}{\sqrt{k}} \sim 2\sqrt{n} - 1. \] Thus, \( x_n \) behaves like: \[ x_n \sim 2\sqrt{n} - 1 - 2\sqrt{n} = -1. \] This suggests that the sequence is bounded and converges to \( -1 \). 
Step 3: Monotonicity. Given that \( x_n \) decreases as \( n \) increases, the sequence is monotonically decreasing. Thus, the correct answer is \( \boxed{(B)} \).