If sn= Sn =(-1)^n  (3+ 1/ 2^n) then find sup Sn and inf Sn by using definition 

Given the sequence \( S_n = (-1)^n \left( 3 + \frac{1}{2^n} \right) \), we are tasked with finding the **supremum** (\(\sup S_n\)) and **infimum** (\(\inf S_n\)) using the formal definition of these concepts.

### Key Observations:
1. The sequence has alternating signs due to the \( (-1)^n \) factor.
   - When \( n \) is **odd**, \( S_n = - \left( 3 + \frac{1}{2^n} \right) \).
   - When \( n \) is **even**, \( S_n = + \left( 3 + \frac{1}{2^n} \right) \).
   
2. As \( n \to \infty \), the term \( \frac{1}{2^n} \) approaches 0, meaning that:
   - \( S_n \to 3 \) for even \( n \).
   - \( S_n \to -3 \) for odd \( n \).

Thus, the sequence approaches values around 3 and -3.

### Behavior of \( S_n \):
- When \( n \) is even, \( S_n = 3 + \frac{1}{2^n} \), which means the terms are slightly larger than 3 but approach 3 as \( n \) increases.
- When \( n \) is odd, \( S_n = - \left( 3 + \frac{1}{2^n} \right) \), so these terms are slightly smaller than \(-3\) and approach \(-3\) as \( n \) increases.

### Supremum and Infimum:
- **Supremum (\(\sup S_n\)):**
  The supremum is the least upper bound of the sequence. Since the even-indexed terms of the sequence are always slightly larger than 3 but get arbitrarily close to 3, the largest value the sequence can take is just above 3.
  
  Therefore, \( \sup S_n = 3 + \frac{1}{2^2} = 3.25 \), which occurs at \( n = 2 \), and no larger value exists.

- **Infimum (\(\inf S_n\)):**
  The infimum is the greatest lower bound of the sequence. The odd-indexed terms are slightly smaller than -3 but approach \(-3\) as \( n \) increases. So, the smallest value the sequence can take is just below \(-3\).

  Therefore, \( \inf S_n = - \left( 3 + \frac{1}{2^1} \right) = -3.5 \), which occurs at \( n = 1 \), and no smaller value exists.

### Conclusion:
- \( \sup S_n = 3.25 \).
- \( \inf S_n = -3.5 \).

Would you like a more detailed explanation of the steps, or do you have any further questions?

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Here are some related questions to explore:

1. How is the behavior of a sequence affected by alternating signs?
2. What is the difference between the supremum and the maximum of a sequence?
3. How do the bounds of a sequence relate to its long-term behavior?
4. What is the impact of the decay term \( \frac{1}{2^n} \) on the sequence?
5. How would you compute limits for different types of sequences?

**Tip:** Always analyze the behavior of even- and odd-indexed terms separately when dealing with alternating sequences.

If sn= Sn =(-1)^n (3+ 1/ 2^n) then find sup Sn and inf Sn by using definition

Explore how to find the supremum and infimum of the alternating sequence Sn = (-1)^n (3 + 1 / 2^n). Understand key concepts like limits, supremum, and infimum in the context of alternating series and sequences. This solution is suitable for undergraduate-level calculus students.

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