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.