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

Answer: (C)

Both the series $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ converge.

Answer 2

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