Select all  statements below which are true about 
an=\cos \left(\pi ^{-n}\right)
Group of answer choices

the sequence is bounded above

the sequence is bounded below

the sequence is bounded

the sequence is non-increasing

the sequence is non-decreasing

the sequence is monotone

the sequence covereges

the sequence diverges

Let's analyze the sequence \( a_n = \cos\left(\pi^{-n}\right) \):

1. **Bounded Above and Below**: 
   - The function \( \cos(x) \) always takes values between \(-1\) and \(1\) for any real number \(x\).
   - Therefore, for all \(n\), \(a_n = \cos(\pi^{-n})\) must also be between \(-1\) and \(1\).
   - So the sequence is bounded above (by 1) and bounded below (by -1).
   - Therefore, **the sequence is bounded**.

2. **Monotonicity**:
   - Consider the argument inside the cosine, \( \pi^{-n} \), which decreases as \(n\) increases.
   - Since \( \cos(x) \) is a periodic function, it's not guaranteed that the values of \( \cos(\pi^{-n}) \) will decrease or increase consistently as \(n\) increases. This function oscillates, meaning the sequence is neither non-increasing nor non-decreasing.
   - Thus, **the sequence is not monotone**.

3. **Convergence or Divergence**:
   - As \( n \to \infty \), \( \pi^{-n} \to 0 \). 
   - Since \( \cos(0) = 1 \), the sequence converges to 1.
   - Hence, **the sequence converges**.

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### Correct True Statements:
- The sequence is bounded above.
- The sequence is bounded below.
- The sequence is bounded.
- The sequence converges.

### False Statements:
- The sequence is non-increasing.
- The sequence is non-decreasing.
- The sequence is monotone.
- The sequence diverges.

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Do you have any questions or would you like further clarification on any part? 

Here are 5 related questions to consider:
1. What does it mean for a sequence to be bounded?
2. How do we formally prove a sequence is convergent?
3. What is the difference between a monotone and oscillating sequence?
4. Can a sequence be bounded and divergent at the same time?
5. How does the periodic nature of cosine affect the behavior of sequences?

**Tip**: When studying sequences, always check if the sequence has a limit as \( n \to \infty \), as this helps in determining whether the sequence converges or diverges.

Select all statements below which are true about an = cos(π^(-n)) Group of answer choices: the sequence is bounded above, the sequence is bounded below, the sequence is bounded, the sequence is non-increasing, the sequence is non-decreasing, the sequence is monotone, the sequence converges, the sequence diverges

Explore the properties of the sequence an = cos(π^(-n)). This detailed analysis covers whether the sequence is bounded, monotonic, and convergent, utilizing key concepts from trigonometry and sequence behavior. Suitable for undergraduate students studying calculus or mathematical analysis.

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