Question

For \( n \in \mathbb{N} \), let \[a_n = \sqrt{n} \sin^2\left(\frac{1}{n}\right) \cos n,\] and \[b_n = \sqrt{n} \sin\left(\frac{1}{n^2}\right) \cos n.\] Then

• A. The series \( \sum_{n=1}^{\infty} a_n \) converges, but the series \( \sum_{n=1}^{\infty} b_n \) does not converge.
• B. The series \( \sum_{n=1}^{\infty} a_n \) does not converge, but the series \( \sum_{n=1}^{\infty} b_n \) converges.
• C. Both the series \( \sum_{n=1}^{\infty} a_n \) and \( \sum_{n=1}^{\infty} b_n \) converge.
• D. Neither the series \( \sum_{n=1}^{\infty} a_n \) nor the series \( \sum_{n=1}^{\infty} b_n \) converges.

Answer 1

The Correct Option is C

The correct option is (C): Both the series \( \sum_{n=1}^{\infty} a_n \) and \( \sum_{n=1}^{\infty} b_n \) converge.