Math 2.0

definition

Monotone sequence

Definition

Let (an)(a_n) be a real sequence. The sequence is increasing if an≤an+1a_n \le a_{n+1} for all n∈Nn \in \mathbb{N}. The sequence is decreasing if an≥an+1a_n \ge a_{n+1} for all n∈Nn \in \mathbb{N}. The sequence is monotone if it is increasing or decreasing. The sequence is strictly increasing if an<an+1a_n < a_{n+1} for all nn, and strictly decreasing if an>an+1a_n > a_{n+1} for all nn.

Remarks

Some texts say non-decreasing for increasing in this sense, and they keep the word increasing for strictly increasing. By induction, (an)(a_n) is increasing if and only if am≤ana_m \le a_n for all m≤nm \le n. This is the definition of a monotone function on N\mathbb{N}. The sequence 1/n1/n is strictly decreasing. The sequence (−1)n(-1)^n is not monotone.

Explanations

L1 human

A monotone sequence always moves in one direction. An increasing sequence never goes down, and a decreasing sequence never goes up.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinition
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  1. Sequence definition
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Sources

  1. Web https://en.wikipedia.org/wiki/Sequence, Increasing and decreasing
  2. Web https://en.wikipedia.org/wiki/Monotonic_function, In calculus and analysis

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