Question

Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit. $$ a_{n}=\frac{3+n^{2}}{1-n} $$

Short answer

The sequence is not bounded, not monotone, and divergent.

Step-by-step solution

Analyze the Sequence

First, we note the given sequence: \( a_n = \frac{3 + n^2}{1 - n} \). This sequence seems to have terms that become quite large as \( n \) increases, suggesting that it may not be bounded.

Check for Monotonicity

To check if \( a_n \) is monotonic, evaluate the derivative or difference: \( a_{n+1} - a_n = \frac{3 + (n+1)^2}{1 - (n+1)} - \frac{3 + n^2}{1 - n} \). Simplifying, if \( a_{n+1} - a_n \) is always positive or negative, the sequence is monotonic. Simplification shows it does not consistently increase or decrease, hence it is not monotonic.

Examine if the Sequence is Bounded

Evaluate \( a_n \) for large values of \( n \), like \( n = 10, 100, \) and \( 1000 \). As \( n \) increases, it appears \( a_n \) grows in magnitude indefinitely (since numerator grows \( n^2 \) times faster than denominator), showing \( a_n \) is not bounded.

Determine Convergence or Divergence

For convergence, the limit \( \lim_{n \to \infty} a_n \) must exist. Simplify the fraction: \( a_n = \frac{n^2(1 + \frac{3}{n^2})}{n(-1 + \frac{1}{n})} \approx -n(1 + \frac{3}{n^2}) \to -\infty \). With the terms approaching infinity negatively, \( a_n \) does not converge, confirming divergence.

Key concepts

Monotonic Sequences

Monotonic sequences are those that consistently increase or decrease as the sequence progresses. This consistent behavior can make calculations easier, especially when dealing with convergence and limits. To determine if a sequence is monotonic, we look at the differences between consecutive terms.- If a sequence is increasing, the difference between consecutive terms will always be positive. - If it's decreasing, the difference will be negative.Taking the sequence in our example, \( a_n = \frac{3 + n^2}{1 - n} \), calculating \( a_{n+1} - a_n \) helps us assess this behavior. Upon simplifying, if there’s no consistency in the sign of this difference, the sequence isn’t monotonic. In this case, the sign changes, suggesting the sequence neither consistently increases nor decreases. Therefore, this sequence is not monotonic.

Bounded Sequences

A sequence is considered bounded if all its terms remain within fixed numerical bounds. These bounds might be either from above (upper bound) or below (lower bound). For a sequence like our example, understanding boundedness involves:- Analyzing large values of \( n \) to see if the magnitude of \( a_n \) keeps growing.- Should \( a_n \) neither explode to \( \infty \) nor collapse to \(-\infty \), it may be bounded.In the given sequence \( a_n = \frac{3 + n^2}{1 - n} \) with its numerator \( n^2 \) overpowering the denominator as \( n \) increases, the terms go beyond any fixed boundary. Therefore, it's clear that this sequence is not bounded, with terms increasing indefinitely in magnitude.

Divergent Sequences

A divergent sequence is one that does not approach a particular limit as the sequence's index goes to infinity. In simple terms, divergence means the sequence either grows without bound, oscillates without settling, or doesn't settle towards any specific point.In considering whether a sequence is convergent or divergent, we look for a limit:- If \( \lim_{n \to \infty} a_n \) yields a finite number, the sequence converges.- If the limit is infinite or doesn't exist, the sequence diverges.For our sequence \( a_n = \frac{3 + n^2}{1 - n} \), simplifying shows terms approach \(-\infty\) as \( n \to \infty \), leading to a conclusion that the sequence diverges as it grows indefinitely towards a negative infinity. This behavior, an indefinite increase in magnitude, confirms its divergence.