Monotone sequence

A sequence \((a_n)\) is increasing if \(a_n\le a_{n+1}\) for all \(n\), and decreasing if \(a_n\ge a_{n+1}\) for all \(n\). When the inequalities are strict we speak of strictly increasing or strictly decreasing sequences. A constant sequence is both increasing and decreasing in the weak sense.

There are three usual ways to check monotonicity of \(a_n\): study the sign of \(a_{n+1}-a_n\), compare the quotient \(\frac{a_{n+1}}{a_n}\) with \(1\) when the terms are positive, or differentiate a function \(f\) with \(a_n=f(n)\). For instance, \(a_n=\frac{n}{n+1}\) satisfies \(a_{n+1}-a_n=\frac1{(n+1)(n+2)}>0\), so it is strictly increasing.

Two mistakes are common. The first is to believe that «monotone» requires every term to change: the sequence \(1,1,2,2,3,3,\dots\) is increasing. The second is to think the first term decides the direction: if \(a_1=-4\) and \(a_{n+1}=\frac{a_n}2\), the terms are negative and the sequence grows toward \(0\). Nor should monotone be confused with bounded: \(a_n=n\) is monotone and unbounded, and \((-1)^n\) is bounded and not monotone.

The sum of increasing sequences is increasing, but the product, difference and quotient need not be. A non-monotone sequence may still converge (like \(\frac{(-1)^n}{n}\)), yet monotonicity gives a very strong convergence criterion in the next concept: the monotone convergence theorem.