Problem 53 - Single Variable Calculus

Question

Determine whether the sequence \(\left\\{a_{n}\right\\}\) is monotonic. Is the sequence bounded? \(a_{n}=3-\frac{1}{n}\)

Step-by-Step Solution

Answer

The sequence \(a_n = 3-\frac{1}{n}\) is monotonic since the difference between consecutive terms, \(a_{n+1} - a_n = \frac{1}{n} - \frac{1}{n+1}\), is positive for all \(n \in \mathbb{N}\). The sequence is also bounded since all terms are greater than \(2\) (lower bound) and converging to \(3\) (upper bound).

Step 1: Define the sequence

The given sequence is defined as \(a_{n}=3-\frac{1}{n}\), where \(n \in \mathbb{N}\).

Step 2: Find the difference between consecutive terms

To determine if the sequence is monotonic, we need to find the difference between consecutive terms, i.e., \(a_{n+1} - a_n\). First, we express \(a_{n+1}\): \(a_{n+1} = 3-\frac{1}{n+1}\) Now, let us calculate the difference between consecutive terms, \(a_{n+1} - a_n\): \(a_{n+1} - a_n = \left(3-\frac{1}{n+1}\right) - \left(3-\frac{1}{n}\right)\)

Step 3: Simplify the difference expression

Next, we need to simplify the expression for the difference we obtained in step 2: \(a_{n+1} - a_n = 3 - \frac{1}{n+1} - 3 + \frac{1}{n}\) \(a_{n+1} - a_n = \frac{1}{n} - \frac{1}{n+1}\)

Step 4: Analyze the sign of the difference expression

Now, we analyze the sign of the simplified difference expression: Since \(n \in \mathbb{N}\), \(n\) will always be a positive integer. Hence, the terms \(\frac{1}{n}\) and \(\frac{1}{n+1}\) are both positive. Since \(n+1 > n\), we have \(\frac{1}{n} > \frac{1}{n+1}\), which implies that their difference is positive: \(\frac{1}{n} - \frac{1}{n+1} > 0\) This indicates that \(a_{n+1} - a_n > 0\) for all \(n \in \mathbb{N}\). The sequence is increasing, so it is monotonic.

Step 5: Determine whether the sequence is bounded

Finally, we need to check if the sequence is bounded. Since \(a_n = 3-\frac{1}{n}\), as \(n\) goes to infinity, the term \(\frac{1}{n}\) goes to zero, and hence, \(a_n\) approaches \(3\). Therefore, the sequence is increasing but converging to \(3\). It is clear that all terms are greater than \(2\) (i.e., \(a_n > 2\) when \(n \geq 1\)). Thus, we have a lower bound \(2\) and an upper bound \(3\), and we conclude that the sequence is monotonic and bounded.

Key Concepts

Bounded Sequences

A sequence is said to be bounded if there exists a real number that serves as an upper bound and another real number that serves as a lower bound for all terms in the sequence. For the sequence given as \(a_n = 3 - \frac{1}{n}\), we can see:

• The sequence approaches 3 as \(n\) becomes very large, indicating that 3 acts as an upper boundary.
• For every term in this sequence when \(n \geq 1\), we have \(a_n = 3 - \frac{1}{n}\) where each term is always greater than 2, establishing 2 as a lower boundary.

Thus, all terms of the sequence are confined within the interval (2, 3). Sequences such as this one, that can be boxed within specific minimum and maximum values, are considered bounded sequences. Recognizing this property helps us to understand their behavior without needing to evaluate every term.

Convergence

In mathematical terms, a sequence is said to converge if its terms approach a specific number as \(n\), the term number, goes infinitely large. The concept of convergence is critically important when considering the long-term behavior of sequences.

• The given sequence \(a_n = 3 - \frac{1}{n}\) converges because as \(n\) gets larger, \(\frac{1}{n}\) becomes smaller and smaller, nearing zero.
• Thus, the value of \(a_n\) becomes increasingly close to 3.

Therefore, sequence convergence here signifies that no matter how far out you go in the sequence, the terms get closer and closer to 3 but never actually exceed it. The destination of this gradual approach represents a fundamental aspect of sequences in studying mathematical limits and continuity.

Natural Numbers

The sequence \(a_n = 3 - \frac{1}{n}\) is defined for natural numbers \(n\). Natural numbers \(\mathbb{N}\) are essentially the set of positive integers starting from 1 upwards (1, 2, 3, ...). These numbers are foundational in mathematics, providing the building blocks for more complex ideas.

• Because the sequence is based on \(n\) being a natural number, \(n\) is always a positive integer, which influences the behavior of the sequence.
• Natural numbers ensure that \(\frac{1}{n}\) is well-defined and remains a positive fraction, allowing \(a_n\) to consistently approximate the limit of 3 as \(n\) grows.

In effect, the use of natural numbers guarantees that the sequence \(a_n = 3 - \frac{1}{n}\) is increasing and helps to establish its monotonic and convergent properties effectively. Understanding natural numbers are crucial for anyone delving into foundational mathematics and realizing how sequences operate.