3.1 Monotone Sequences and Their Properties

Monotone sequences are like predictable friends - they always move in the same direction. They come in four flavors: increasing, decreasing, non-increasing, and non-decreasing. Understanding these sequences helps us grasp how they behave and where they're headed.

The cool thing about monotone sequences is that if they're bounded, they always converge. This means they settle down to a specific value as we keep going. It's a powerful tool for proving convergence without actually finding the exact limit.

Monotone sequences and their types

Definition and classification of monotone sequences

• A sequence $\{a_n\}$ is monotone if it maintains a consistent order relation between consecutive terms for all $n \in \mathbb{N}$
  • The four types of monotone sequences are increasing, decreasing, non-increasing, and non-decreasing
  • Strictly monotone sequences ($a_n < a_{n+1}$ or $a_n > a_{n+1}$) are either increasing or decreasing
  • Non-strictly monotone sequences ($a_n \leq a_{n+1}$ or $a_n \geq a_{n+1}$) are either non-increasing or non-decreasing

Definitions and examples of each type of monotone sequence

• A sequence $\{a_n\}$ is increasing if $a_n < a_{n+1}$ for all $n \in \mathbb{N}$
  • Example: the sequence $\{1, 2, 3, 4, ...\}$ is increasing
  • Example: the sequence $\{1/n\}$ for $n \in \mathbb{N}$ is increasing
• A sequence $\{a_n\}$ is decreasing if $a_n > a_{n+1}$ for all $n \in \mathbb{N}$
  • Example: the sequence $\{10, 9, 8, 7, ...\}$ is decreasing
  • Example: the sequence $\{1/n^2\}$ for $n \in \mathbb{N}$ is decreasing
• A sequence $\{a_n\}$ is non-increasing if $a_n \geq a_{n+1}$ for all $n \in \mathbb{N}$
  • Example: the sequence $\{5, 5, 4, 4, 3, 3, ...\}$ is non-increasing
  • Example: the sequence $\{1 + 1/n\}$ for $n \in \mathbb{N}$ is non-increasing
• A sequence $\{a_n\}$ is non-decreasing if $a_n \leq a_{n+1}$ for all $n \in \mathbb{N}$
  • Example: the sequence $\{1, 1, 2, 2, 3, 3, ...\}$ is non-decreasing
  • Example: the sequence $\{1 - 1/n\}$ for $n \in \mathbb{N}$ is non-decreasing

Properties of monotone sequences

Boundedness of monotone sequences

• Every monotone sequence is bounded
  • An increasing sequence is bounded below by its first term and above by any subsequent term
    • Example: the sequence $\{1, 2, 3, 4, ...\}$ is bounded below by 1 and above by any term in the sequence
  • A decreasing sequence is bounded above by its first term and below by any subsequent term
    • Example: the sequence $\{10, 9, 8, 7, ...\}$ is bounded above by 10 and below by any term in the sequence
• The boundedness of monotone sequences is a crucial property for proving convergence

Convergence of bounded monotone sequences

• Every bounded monotone sequence converges
  • For an increasing sequence $\{a_n\}$ bounded above by $M$, the limit $L = \sup\{a_n: n \in \mathbb{N}\}$ exists, and $\lim_{n \to \infty} a_n = L$
    • Example: the sequence $\{1 - 1/n\}$ for $n \in \mathbb{N}$ is increasing and bounded above by 1, so it converges to 1
  • For a decreasing sequence $\{a_n\}$ bounded below by $m$, the limit $L = \inf\{a_n: n \in \mathbb{N}\}$ exists, and $\lim_{n \to \infty} a_n = L$
    • Example: the sequence $\{1/n\}$ for $n \in \mathbb{N}$ is decreasing and bounded below by 0, so it converges to 0
• The limit of a convergent monotone sequence is unique

Monotonicity and limits of sequences

Determining monotonicity and finding limits

• To determine the monotonicity of a sequence, compare consecutive terms using the definitions of increasing, decreasing, non-increasing, and non-decreasing sequences
  • Example: for the sequence $\{a_n\} = \{1/n\}$, compare $a_n$ and $a_{n+1}$ to show that $1/n > 1/(n+1)$ for all $n \in \mathbb{N}$, proving that the sequence is decreasing
• If a sequence is monotone and bounded, it converges to a limit $L$
  • Example: the sequence $\{1/n\}$ is decreasing and bounded below by 0, so it converges to a limit $L = 0$
• To find the limit of a monotone sequence, use algebraic manipulation, the Squeeze Theorem, or the definition of the limit
  • Example: to find the limit of $\{1/n\}$, use the Squeeze Theorem with the sequences $\{0\}$ and $\{1/n\}$ to show that $\lim_{n \to \infty} 1/n = 0$

Divergence of unbounded monotone sequences

• If a monotone sequence is unbounded, it diverges to either $\infty$ or $-\infty$, depending on whether it is increasing or decreasing, respectively
  • Example: the sequence $\{n\}$ is increasing and unbounded, so it diverges to $\infty$
  • Example: the sequence $\{-n\}$ is decreasing and unbounded, so it diverges to $-\infty$

Monotone Convergence Theorem

Statement and application of the theorem

• The Monotone Convergence Theorem states that every bounded monotone sequence converges
• To apply the Monotone Convergence Theorem:
  1. Prove that the sequence is monotone (increasing, decreasing, non-increasing, or non-decreasing)
  2. Prove that the sequence is bounded (find a lower or upper bound, depending on the monotonicity)
  3. Conclude that the sequence converges by the Monotone Convergence Theorem
• The Monotone Convergence Theorem can be used to prove the convergence of sequences without explicitly finding the limit
  • Example: to prove that the sequence $\{(1 + 1/n)^n\}$ converges, show that it is increasing and bounded above by $e$, then apply the Monotone Convergence Theorem

Examples of using the theorem to prove convergence

• Example: prove that the sequence $\{a_n\} = \{1 - 1/n^2\}$ converges
  1. Show that $\{a_n\}$ is increasing: $a_{n+1} - a_n = 1/(n^2(n+1)^2) > 0$ for all $n \in \mathbb{N}$
  2. Show that $\{a_n\}$ is bounded above by 1: $a_n = 1 - 1/n^2 < 1$ for all $n \in \mathbb{N}$
  3. Apply the Monotone Convergence Theorem to conclude that $\{a_n\}$ converges
• Example: prove that the sequence $\{b_n\} = \{n/(n+1)\}$ converges
  1. Show that $\{b_n\}$ is increasing: $b_{n+1} - b_n = 1/((n+1)(n+2)) > 0$ for all $n \in \mathbb{N}$
  2. Show that $\{b_n\}$ is bounded above by 1: $b_n = n/(n+1) < 1$ for all $n \in \mathbb{N}$
  3. Apply the Monotone Convergence Theorem to conclude that $\{b_n\}$ converges