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.