a^n = [n^2] / [1+n^6]. Is this sequence bounded or unbounded? Is this sequence monotonic or not monotonic? Does this sequence converge or diverge? If the sequence does converge, to what value?

We are given the sequence:

\[
a_n = \frac{n^2}{1 + n^6}
\]

We need to analyze whether this sequence is **bounded** or **unbounded**, **monotonic** or **non-monotonic**, and whether it **converges** or **diverges**. If it converges, we also need to find the value to which it converges.

### 1. **Boundedness:**
To check if the sequence is bounded, we need to determine if there exists a real number \( M \) such that \( |a_n| \leq M \) for all \( n \).

- For large \( n \), the denominator \( 1 + n^6 \) grows much faster than the numerator \( n^2 \), so the value of the fraction \( a_n \) tends to decrease as \( n \) increases.

- Let's check the limit as \( n \to \infty \):
  
  \[
  \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{n^2}{1 + n^6}
  \]
  To compute this limit, divide both the numerator and the denominator by \( n^6 \):
  \[
  \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{n^2 / n^6}{(1 + n^6) / n^6} = \lim_{n \to \infty} \frac{1 / n^4}{1 + 1 / n^6} = 0
  \]
  Hence, \( a_n \to 0 \) as \( n \to \infty \).

  For small values of \( n \), the sequence \( a_n \) is non-negative and reaches a maximum for small values of \( n \). Therefore, the sequence is **bounded**.

### 2. **Monotonicity:**
To check whether the sequence is monotonic, we need to check the difference \( a_{n+1} - a_n \) or the ratio \( \frac{a_{n+1}}{a_n} \).

- The general form of the sequence is:
  \[
  a_n = \frac{n^2}{1 + n^6}
  \]

- To determine if it is monotonic, we need to check if \( a_n \) is increasing or decreasing. Consider the derivative of the continuous analogue \( f(x) = \frac{x^2}{1 + x^6} \):
  
  \[
  f'(x) = \frac{2x(1 + x^6) - x^2(6x^5)}{(1 + x^6)^2} = \frac{2x - 6x^7}{(1 + x^6)^2}
  \]
  Factorizing:
  \[
  f'(x) = \frac{2x(1 - 3x^6)}{(1 + x^6)^2}
  \]
  The sign of \( f'(x) \) depends on \( 1 - 3x^6 \). For small values of \( x \), \( f'(x) > 0 \), so the sequence is increasing for small \( x \). However, for large \( x \), \( f'(x) < 0 \), meaning the sequence is decreasing for large \( x \). Hence, the sequence is **not monotonic**.

### 3. **Convergence:**
As we computed earlier:
\[
\lim_{n \to \infty} a_n = 0
\]
Thus, the sequence **converges** to 0.

### Summary:
- The sequence is **bounded**.
- The sequence is **not monotonic**.
- The sequence **converges** to **0**.

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Would you like more details on any of these steps or concepts?

### 5 Related Questions:
1. How do you prove a sequence is bounded mathematically?
2. What is the significance of monotonicity in sequence convergence?
3. How can you find the exact turning point where a sequence stops being monotonic?
4. Can bounded and non-monotonic sequences still converge?
5. How does the growth rate of terms in a sequence affect its convergence?

**Tip**: When checking for boundedness, finding the limit of the sequence as \( n \to \infty \) can provide quick insights into its long-term behavior.

This detailed solution explores whether the sequence a_n = n^2 / (1 + n^6) is bounded, monotonic, and whether it converges or diverges. Learn key concepts such as sequence boundedness, monotonicity, and how the sequence converges to 0 for large values of n. Suitable for students in undergraduate calculus.

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