Determining Bounded and Monotonic Sequences

Introduction

Key Concepts

Sequences: An Overview

A sequence is an ordered list of numbers that follow a specific pattern or rule. Formally, a sequence is a function whose domain is the set of natural numbers. Sequences can be finite or infinite, with infinite sequences being the primary focus in calculus due to their relevance in defining series and understanding limits.

Bounded Sequences

A sequence $\{a_n\}$ is said to be bounded if there exists a real number $M$ such that for all $n \in \mathbb{N}$, the absolute value of $a_n$ does not exceed $M$. Mathematically, this is expressed as: $$|a_n| \leq M \quad \text{for all } n \in \mathbb{N}$$ Boundedness ensures that the sequence does not diverge to infinity or negative infinity. It is a crucial property when applying the Bolzano-Weierstrass Theorem, which guarantees that every bounded sequence has at least one convergent subsequence. **Types of Bounded Sequences:** 1. **Upper Bounded:** There exists an $M$ such that $a_n \leq M$ for all $n$. 2. **Lower Bounded:** There exists an $m$ such that $a_n \geq m$ for all $n$. 3. **Absolutely Bounded:** There exists an $M$ where $|a_n| \leq M$ for all $n$. **Examples:** - The sequence $a_n = \frac{1}{n}$ is bounded since $0 < a_n \leq 1$ for all $n$. - The sequence $a_n = (-1)^n$ is bounded as it oscillates between -1 and 1.

Monotonic Sequences

A sequence $\{a_n\}$ is monotonic if it is entirely non-increasing or non-decreasing. Monotonicity provides insight into the convergence behavior of sequences. **Types of Monotonic Sequences:** 1. **Monotonically Increasing:** For all $n \in \mathbb{N}$, $a_{n+1} \geq a_n$. 2. **Strictly Increasing:** For all $n \in \mathbb{N}$, $a_{n+1} > a_n$. 3. **Monotonically Decreasing:** For all $n \in \mathbb{N}$, $a_{n+1} \leq a_n$. 4. **Strictly Decreasing:** For all $n \in \mathbb{N}$, $a_{n+1} < a_n$. Monotonic sequences are significant because every bounded monotonic sequence is convergent, a result derived from the Monotone Convergence Theorem. **Examples:** - The sequence $a_n = n$ is monotonically increasing. - The sequence $a_n = \frac{1}{n}$ is monotonically decreasing.

Determining Boundedness

To determine if a sequence is bounded, identify constants that serve as upper and lower bounds. For instance, consider the sequence $a_n = \sin\left(\frac{n\pi}{2}\right)$. Since the sine function oscillates between -1 and 1, the sequence is bounded with $M = 1$. **Steps to Determine Boundedness:** 1. **Identify the General Term:** Express the sequence in terms of $n$. 2. **Analyze the Behavior:** Consider the limit as $n \to \infty$ and the range of the terms. 3. **Find Bounds:** Determine constants that the sequence does not exceed in magnitude.

Determining Monotonicity

To ascertain if a sequence is monotonic, evaluate the difference $a_{n+1} - a_n$. **Steps to Determine Monotonicity:** 1. **Compute the Difference:** Calculate $a_{n+1} - a_n$. 2. **Analyze the Sign:** - If $a_{n+1} - a_n \geq 0$ for all $n$, the sequence is monotonically increasing. - If $a_{n+1} - a_n \leq 0$ for all $n$, the sequence is monotonically decreasing. 3. **Consider Strict Inequalities:** If the inequalities are strict, the sequence is strictly monotonic. **Example:** Consider $a_n = \frac{n}{n+1}$. Compute: $$a_{n+1} - a_n = \frac{n+1}{n+2} - \frac{n}{n+1} = \frac{(n+1)^2 - n(n+2)}{(n+1)(n+2)} = \frac{n^2 + 2n + 1 - n^2 - 2n}{(n+1)(n+2)} = \frac{1}{(n+1)(n+2)} > 0$$ Since $a_{n+1} - a_n > 0$ for all $n$, the sequence is monotonically increasing.

Convergence of Bounded and Monotonic Sequences

The Monotone Convergence Theorem states that every bounded monotonic sequence is convergent. This is pivotal in calculus, especially when evaluating the limits of sequences and series. **Theorem (Monotone Convergence):** If a sequence $\{a_n\}$ is monotonic and bounded, then it is convergent. **Proof Outline:** 1. **Monotonically Increasing:** If the sequence is increasing and bounded above, it converges to its least upper bound (supremum). 2. **Monotonically Decreasing:** If the sequence is decreasing and bounded below, it converges to its greatest lower bound (infimum). **Applications:** - **Evaluating Limits:** Determining the limit of a sequence by establishing its boundedness and monotonicity. - **Series Convergence:** Assessing the convergence of infinite series by analyzing the boundedness and behavior of their partial sums.

Examples and Applications

**Example 1: Bounded Monotonic Sequence** Consider $a_n = \frac{1}{n}$. - **Boundedness:** $0 < a_n \leq 1$. - **Monotonicity:** Monotonically decreasing. - **Conclusion:** By the Monotone Convergence Theorem, $a_n$ converges to 0. **Example 2: Unbounded Sequence** Consider $a_n = n$. - **Boundedness:** Not bounded above. - **Monotonicity:** Monotonically increasing. - **Conclusion:** Since the sequence is unbounded, it diverges to infinity. **Example 3: Oscillating Sequence** Consider $a_n = (-1)^n$. - **Boundedness:** $-1 \leq a_n \leq 1$. - **Monotonicity:** Not monotonic as it oscillates. - **Conclusion:** The sequence does not converge.

Common Misconceptions

1. **All Bounded Sequences Converge:** This is false; boundedness alone does not guarantee convergence without monotonicity. 2. **Monotonic Sequences Are Always Bounded:** Not necessarily. A sequence can be monotonic but unbounded, leading to divergence. 3. **Strict Monotonicity Implies Boundedness:** Strict monotonicity only affects the behavior of the sequence but does not ensure boundedness.

Advanced Topics

**Cauchy Sequences:** A sequence $\{a_n\}$ is a Cauchy sequence if for every $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that for all $m, n > N$, $|a_n - a_m| < \epsilon$. In complete metric spaces like $\mathbb{R}$, every Cauchy sequence is convergent. **Boundedness in Higher Dimensions:** While boundedness in $\mathbb{R}$ deals with upper and lower limits along a line, in higher-dimensional spaces, boundedness involves containment within a finite region, such as a disk or sphere. **Supremum and Infimum:** The supremum (least upper bound) and infimum (greatest lower bound) are pivotal in defining the limits of bounded sequences, providing the exact values to which monotonic sequences converge.

Comparison Table

Aspect	Bounded Sequences	Monotonic Sequences
Definition	Sequences confined within upper and/or lower limits	Sequences that are entirely non-increasing or non-decreasing
Convergence Guarantee	No, unless combined with monotonicity	Yes, if the sequence is also bounded
Examples	$a_n = \frac{1}{n}$, $a_n = (-1)^n$	$a_n = n$, $a_n = \frac{1}{n}$
Applications	Bolzano-Weierstrass Theorem, determining subsequences	Monotone Convergence Theorem, evaluating limits
Key Properties	Containness within bounds prevents divergence	Single direction of movement aids in convergence proofs

Summary and Key Takeaways