Question

Let \(\{x_n\}_{n \geq 1}\) be a sequence of positive real numbers such that the series \(\sum_{n=1}^{\infty} x_n\) converges. Which of the following statements is (are) always TRUE?

• A. The series \(\sum_{n=1}^{\infty} \sqrt{x_n x_{n+1}}\) converges
• B. \(\lim_{n \to \infty} x_n = 0\)
• C. The series \(\sum_{n=1}^{\infty} x_n^2\) converges
• D. The series \(\sum_{n=1}^{\infty} \frac{\sqrt{x_n}}{1 + \sqrt{x_n}}\) converges

Hint:

For a convergent series \(\sum_{n=1}^{\infty} x_n\), the terms \(x_n\) must approach zero as \(n \to \infty\).

Answer 1

The Correct Option is A, C

Step 1: Understanding the convergence of the series.
For the series \(\sum_{n=1}^{\infty} x_n\) to converge, it is necessary that \(x_n \to 0\) as \(n \to \infty\). This is a fundamental property of converging series of positive terms. Therefore, \(\lim_{n \to \infty} x_n = 0\) must hold.
Step 2: Analyzing the options.
(A) The series \(\sum_{n=1}^{\infty} \sqrt{x_n x_{n+1}}\) converges: This is not always true. While the terms \(x_n\) may converge, the product \(\sqrt{x_n x_{n+1}}\) does not necessarily behave in a way that guarantees the convergence of the series.
(B) \(\lim_{n \to \infty} x_n = 0\): This is true, as discussed in Step 1.
(C) The series \(\sum_{n=1}^{\infty} x_n^2\) converges: This is not guaranteed by the convergence of \(\sum_{n=1}^{\infty} x_n\), as squaring the terms may not ensure convergence.
(D) The series \(\sum_{n=1}^{\infty} \frac{\sqrt{x_n}}{1 + \sqrt{x_n}}\) converges: This is not guaranteed either, as the rate at which \(x_n\) approaches zero affects the convergence of this series.
Step 3: Conclusion.
The correct answer is \((B)\), since for a series to converge, it is necessary that \(x_n \to 0\).