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?
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?
Updated On: Oct 1, 2024
- the series \( \sum_{n=1}^{\infty} a_n \) converges
- the series \( \sum_{n=1}^{\infty} b_n \) converges
- the series \( \sum_{n=1}^{\infty} a_n \) converges but the series \( \sum_{n=1}^{\infty} b_n \) does NOT converge
- neither the series \( \sum_{n=1}^{\infty} a_n \) nor the series \( \sum_{n=1}^{\infty} b_n \) converges
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The Correct Option is A, B
Solution and Explanation
The correct option is (A): the series \( \sum_{n=1}^{\infty} a_n \) converges ,(B): the series \( \sum_{n=1}^{\infty} b_n \) converges
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