Question

For \( n \in \mathbb{N} \), define \( x_n \) and \( y_n \) by \[ x_n = (-1)^n \cos \frac{1}{n} \quad \text{and} \quad y_n = \sum_{k=1}^{n} \frac{1}{n + k}. \] Then, which one of the following is TRUE?

• A. \( \sum_{n=1}^{\infty} x_n \) converges, and \( \sum_{n=1}^{\infty} y_n \) does NOT converge
• B. \( \sum_{n=1}^{\infty} x_n \) does NOT converge, and \( \sum_{n=1}^{\infty} y_n \) converges
• C. \( \sum_{n=1}^{\infty} x_n \) converges, and \( \sum_{n=1}^{\infty} y_n \) converges
• D. \( \sum_{n=1}^{\infty} x_n \) does NOT converge, and \( \sum_{n=1}^{\infty} y_n \) does NOT converge

Hint:

When dealing with series, check if the terms tend to zero and use tests like the Alternating Series Test for convergence. For non-decreasing terms, check for divergence.

Answer 1

The Correct Option is D

Step 1: Analyze \( \sum_{n=1}^{\infty} x_n \).
The sequence \( x_n = (-1)^n \cos \frac{1}{n} \) oscillates, and since \( \cos \frac{1}{n} \to 1 \) as \( n \to \infty \), the series \( \sum_{n=1}^{\infty} x_n \) behaves like the alternating series \( \sum_{n=1}^{\infty} (-1)^n \), which does NOT converge because the terms do not tend to zero as needed for convergence. 
Step 2: Analyze \( \sum_{n=1}^{\infty} y_n \).
The sequence \( y_n = \sum_{k=1}^{n} \frac{1}{n + k} \) behaves like a harmonic series, which does NOT converge because each term in the series does not tend to zero fast enough for convergence. Hence, the series \( \sum_{n=1}^{\infty} y_n \) also does NOT converge. 
Final Answer: \[ \boxed{ \sum_{n=1}^{\infty} x_n \text{ does NOT converge, and } \sum_{n=1}^{\infty} y_n \text{ does NOT converge.} } \]