Question

Let $a_1 = 1$, $a_{n+1} = a_n \left( \frac{\sqrt{n} + \sin n}{n} \right)$, and $b_n = a_n^2$ for all $n \in \mathbb{N}$. Then which of the following statements is/are correct?

• A. the series $\sum_{n=1}^{\infty} a_n$ converges
• B. the series $\sum_{n=1}^{\infty} b_n$ converges
• C. the series $\sum_{n=1}^{\infty} a_n$ converges but the series $\sum_{n=1}^{\infty} b_n$ does NOT converge
• D. neither the series $\sum_{n=1}^{\infty} a_n$ nor the series $\sum_{n=1}^{\infty} b_n$ converges

Answer 1

Answer: (A), (B)

the series $\sum_{n=1}^{\infty} a_n$ converges

Answer 2

The correct option is (A): the series $\sum_{n=1}^{\infty} a_n$ converges ,(B): the series $\sum_{n=1}^{\infty} b_n$ converges