Question:
Let \( a_n = \left\{ \begin{array}{ll}
2 + (-1)^{n \frac{n}{2^n} & \text{if } n \text{ is odd}
1 + \frac{n}{2^n} & \text{if } n \text{ is even}, \end{array} \right. \ n \in \mathbb{N}.}
Then which one of the following is TRUE?}
Let \( a_n = \left\{ \begin{array}{ll}
2 + (-1)^{n \frac{n}{2^n} & \text{if } n \text{ is odd}
1 + \frac{n}{2^n} & \text{if } n \text{ is even}, \end{array} \right. \ n \in \mathbb{N}.}
Then which one of the following is TRUE?}
1 + \frac{n}{2^n} & \text{if } n \text{ is even}, \end{array} \right. \ n \in \mathbb{N}.}
Then which one of the following is TRUE?}
Show Hint
For sequences with alternating behavior, the \( \lim \inf \) and \( \lim \sup \) represent the smallest and largest accumulation points of the sequence.
Updated On: Dec 11, 2025
- \( \sup \{a_n \mid n \in \mathbb{N} \} = 3 \) and \( \inf \{a_n \mid n \in \mathbb{N} \} = 1 \)
- \( \lim \inf (a_n) = \lim \sup (a_n) = \frac{3}{2} \)
- \( \sup \{a_n \mid n \in \mathbb{N} \} = 3 \) and \( \inf \{a_n \mid n \in \mathbb{N} \} = 1 \)
- \( \lim \inf (a_n) = 1 \) and \( \lim \sup (a_n) = 3 \)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the problem.
We are given the sequence \( a_n \), and we need to find the supremum and infimum of the sequence, as well as the limits of the infimum and supremum.
Step 2: Analyze the behavior of the sequence.
For odd \( n \), the sequence behaves as \( a_n = 2 + \frac{n}{2^n} \), which converges to 2. For even \( n \), the sequence behaves as \( a_n = 1 + \frac{n}{2^n} \), which also converges to 1. Thus, the sequence oscillates between values near 2 and 1.
Step 3: Conclusion.
The correct answer is (D) \( \lim \inf (a_n) = 1 \) and \( \lim \sup (a_n) = 3 \).
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