Question

Determine whether the sequence is monotonically increasing or decreasing. If it is not, determine if there is an \(m\) such that it is monotonic for all \(n \geq m\). $$\left\\{a_{n}\right\\}=\left\\{\frac{n^{2}-6 n+9}{n}\right\\}$$

Short answer

The sequence becomes monotonically increasing for all \( n \geq 4 \).

Step-by-step solution

Simplify the Sequence

Given the sequence \( \{a_n\} = \{\frac{n^2 - 6n + 9}{n}\} \). Simplify this by dividing each term in the numerator by \(n\): \[ a_n = \frac{n^2}{n} - \frac{6n}{n} + \frac{9}{n} = n - 6 + \frac{9}{n} \]. So, \( a_n = n - 6 + \frac{9}{n} \).

Determine Behavior of the Sequence Components

For \( a_n = n - 6 + \frac{9}{n} \), consider each part separately:- \( n - 6 \) is a linear function that increases as \( n \) increases.- \( \frac{9}{n} \) is a decreasing function as \( n \) increases because the numerator is constant and the denominator increases.Thus, the sequence is made of an increasing part and a decreasing part.

Analyze Limits for Large \( n \)

To determine the sequence behavior as \( n \to \infty \), analyze the limit:\[ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(n - 6 + \frac{9}{n}\right) \].The limit is:- \( n - 6 \to \infty \) as \( n \to \infty \).- \( \frac{9}{n} \to 0 \) as \( n \to \infty \).Thus, \( \lim_{n \to \infty} a_n = \infty \).

Search for Monotonicity after a Point

Check if the sequence becomes monotonic for some \( n \geq m \). Compute the derivative of \( a_n \) as a function \( f(n) = n - 6 + \frac{9}{n} \):- The derivative \( f'(n) = 1 - \frac{9}{n^2} \).For \( f'(n) > 0 \):\( 1 - \frac{9}{n^2} > 0 \Rightarrow n^2 > 9 \Rightarrow n > 3 \).Thus, for \( n \geq 4 \), the sequence is increasing, indicating it is monotonic for all \( n \geq 4 \).

Key concepts

Sequence Analysis

Sequence analysis involves examining sequences to determine their properties such as monotonicity, limit, and divergence. In simpler words, it helps us understand how a sequence behaves as we move through its terms.

• A sequence is said to be monotonic if it is either entirely non-increasing or non-decreasing as the sequence progresses.
• Identifying if a sequence is monotonic involves checking its terms systematically or using calculus techniques like differentiation to examine behavior over an interval.

In the given exercise, the sequence is initially neither fully increasing nor decreasing because it involves two parts with opposite behaviors: one increasing and one decreasing. Breaking it down reveals the changes in the individual components:
- The linear part, \(n - 6\), which grows larger as \(n\) increases.
- The fraction \(\frac{9}{n}\), which diminishes as \(n\) grows.
Analysis of these parts helps us predict the sequence's behavior and determine where, if at all, it becomes monotonic.

Limit of a Sequence

The limit of a sequence is a fundamental concept in calculus, describing the value that a sequence approaches as the index \(n\) goes to infinity.
Understanding the limit is essential to comprehend long-term behavior of sequences. For the sequence \(a_n = n - 6 + \frac{9}{n}\), we analyze its limit as \(n\) goes to \(\infty\).

• The term \(n - 6\) approaches \(\infty\) since \(n\) increases indefinitely.
• Meanwhile, \(\frac{9}{n}\) gets closer to \(0\) because the denominator keeps growing.

Consequently, \(\lim_{n \to \infty} a_n = \infty\), indicating that the sequence becomes unbounded as \(n\) grows larger. Recognizing this limit helps us understand the sequence's end behavior, confirming that it tends toward infinity rather than settling at a finite value or oscillating indefinitely.

Calculus

Calculus provides us with powerful tools to probe the properties of sequences, particularly through differentiation, which can identify monotonic behavior.
In the context of sequences, we treat \(a_n\) as a continuous function \(f(n)\) to facilitate analysis using calculus.

• Differentiation helps us derive \(f'(n) = 1 - \frac{9}{n^2}\), a formula showcasing the rate of change of our function with respect to \(n\).
• By setting \(f'(n) > 0\), we solve \(1 - \frac{9}{n^2} > 0\), leading to \(n^2 > 9\) and \(n > 3\).

The differentiation reveals that for \(n \geq 4\), the function becomes increasing, thus defining a point beyond which the sequence is consistently monotonic. Such analytical techniques bridge gaps in understanding natural progression and shifts in sequence behavior. Calculus is indispensable for exploring deeper insights into sequences and their long-term tendencies.