Increasing, decreasing and constant sequences

Definition

A sequence (u_n) is increasing if, for every natural number n, u(n+1) ≥ u_n. It is strictly increasing if u(n+1) > u_n. Similarly, it is decreasing if u(n+1) ≤ u_n, and strictly decreasing if u(n+1) < u_n. If u(n+1) = u_n for all n, the sequence is constant.

A sequence that is increasing or decreasing (in the broad or strict sense) is said to be monotonic. Note: many sequences are neither increasing nor decreasing (for example, an alternating sequence); we then say that they are not monotonic.

Examples

• u_n = n^2: the terms 0, 1, 4, 9, 16, … increase constantly -> strictly increasing sequence.
• u_n = 1/(n+1): the terms 1, 1/2, 1/3, 1/4, … decrease -> strictly decreasing sequence.
• u_n = (-1)^n: the terms are 1, -1, 1, -1, … -> non-monotonic sequence.

Summary table

Condition for all n	Name
u(n+1) > u_n	strictly increasing
u(n+1) >= u_n	increasing (in the broad sense)
u(n+1) < u_n	strictly decreasing
u(n+1) <= u_n	decreasing (in the broad sense)
u(n+1) = u_n	constant

A classic pitfall

Looking only at the first two or three terms is never enough to draw a conclusion: one must prove the inequality for all integers n, generally by algebraic calculation (see the next lesson).