Monotonic Sequence

TL;DR

• Terms of a monotonic sequence consistently get larger or consistently get smaller as the sequence progresses.
• Common examples: the natural numbers 1, 2, 3, … and the squares 1, 4, 9, …
• Monotonicity is used in calculus and analysis to study properties like differentiability, continuity, and convergence.

Definition

A monotonic sequence is a sequence in which the terms either strictly increase or strictly decrease. This means that the terms of the sequence either consistently get larger or consistently get smaller as the sequence progresses.

Explanation

A monotonically increasing sequence has terms that strictly increase at each step; a monotonically decreasing sequence has terms that strictly decrease at each step. Monotonic sequences are important in areas such as calculus and analysis.

In analysis, a sequence {x_n} is said to converge to a limit L if, for every ε>0, there exists an integer N such that

$$|x_n - L| < \varepsilon \quad\text{for all } n \ge N.$$

If a sequence {x_n} is monotonically increasing and bounded above, then it is convergent. Similarly, if a sequence {x_n} is monotonically decreasing and bounded below, then it is also convergent.

Examples

Natural numbers

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … In this sequence, the terms strictly increase by 1 each time, so the sequence is monotonically increasing.

Squares of natural numbers

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, … In this sequence, the terms are obtained by squaring the natural numbers, so they strictly increase as the natural numbers increase; thus the sequence is monotonically increasing.

Use cases

• Calculus: The source states that if a function f is monotonically increasing on an interval [a,b], then it is differentiable on that interval. Similarly, if f is monotonically decreasing on [a,b], then it is differentiable on that interval. The derivative can then be used to study the function’s behavior.
• Analysis: Monotonic sequences are used to define and study concepts such as convergence and divergence, and to determine when sequences approach limits.

Related terms

• Monotonically increasing
• Monotonically decreasing
• Convergence
• Divergence
• Bounded above
• Bounded below
• Limit
• Derivative
• Continuity
• Calculus
• Analysis
• Sequence