Monotone sequence

Definition

A monotone sequence is a sequence of numbers that is either entirely non-increasing or non-decreasing. In other words, the terms either consistently increase or consistently decrease.

5 Must Know Facts For Your Next Test

1. A sequence $\{a_n\}$ is monotone increasing if $a_{n+1} \geq a_n$ for all $n$.
2. A sequence $\{a_n\}$ is monotone decreasing if $a_{n+1} \leq a_n$ for all $n$.
3. Monotone sequences are always bounded by their initial term and any subsequent terms.
4. The Monotone Convergence Theorem states that every bounded monotone sequence converges.
5. To prove a sequence is monotone, you can show that the difference between consecutive terms always has the same sign.

Review Questions

• What conditions must be met for a sequence to be considered monotone?
• How does the Monotone Convergence Theorem relate to bounded sequences?
• Give an example of both a monotone increasing and a monotone decreasing sequence.

Related terms

Bounded Sequence: A sequence $\{a_n\}$ is bounded if there exists a real number $M$ such that $|a_n| \leq M$ for all $n$.

Convergent Sequence: A sequence $\{a_n\}$ converges to a limit $L$ if for every $\epsilon > 0$, there exists an integer $N$ such that for all $n \geq N$, $|a_n - L| < \epsilon$.

Subsequence: A subsequence is derived from another sequence by deleting some or none of its elements without changing the order of the remaining elements.