$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$

Monotonic sequences

Definition

We say that a sequence $(a_n)$ is an increasing sequence if $a_n\leq a_{n+1}$ for all $n\in \N$.

We say $(a_n)$ is a strictly increasing sequence if $a_n\lt a_{n+1}$ for all $n\in \N$.

Examples

1. The sequence $a_n=n$ is a strictly increasing sequence as $n The sequence $a_n=3-1/n$ is a strictly increasing sequence.
2. The sequence $a_n=(-1)^n$ is not an increasing sequence.
3. The sequence $1,1,2,2,3,3,4,4,5,5,\dots $ is an increasing sequence. It is not strictly increasing since for example $a_1=1\not< 1=a_2$. That is, this collides with our usual understanding of increasing, an increasing sequence does not have to increase at every step!
4. In fact, and this is the really counter-intuitive bit so pay particular attention, the sequence $2,2,2,2,2,\dots $ is an increasing sequence! Note that is fits the definition in that $a_n=2\leq 2 =a_{n+1}$ for all $n\in \N$. Also note that it is clearly not strictly increasing.

How to think like a mathematician
The last example demonstrates the importance of being precise about definitions. Intuitively, we think of increasing as ‘always getting bigger’. This is wrong according to our definition.

Exercises

1. Is $a_n=\cos(n)$ an increasing sequence? Is it strictly increasing?

Once we have understood what increasing means it is a simple matter to define decreasing.

Definition
We say that a sequence $(a_n)$ is a decreasing sequence if $a_{n+1}\leq a_{n}$ for all $n\in \N$.

We say $(a_n)$ is a strictly decreasing sequence if $a_{n+1}\lt a_{n}$ for all $n\in \N$.

Examples

1. The sequence $1/n$ is a strictly decreasing sequence.
2. The sequence $-1,-1,-2,-2,-3,-3,\dots $ is a decreasing but not strictly decreasing sequence.
3. If the sequence $a_n$ is (strictly) increasing, then $-a_n$ is (strictly) increasing and vice versa.
4. The sequence $a_n=n$ is not decreasing. (This is because it does not satisfy the definition of decreasing not because it is increasing. The idea that increasing and decreasing are strict opposites is dealt with in the next surprising example.)
5. This is the confusing one: The sequence $2,2,2,2,2,\dots $ is a decreasing sequence!
   It fits the definition of decreasing and the definition of increasing. That is, a sequence can be increasing and decreasing.

Exercises

1. Show that for $n\geq 3$, $n^{1/n}$ is a decreasing sequence.
2. Is $\sin (\pi /2n) $ a decreasing sequence? Explain your reasons.

Definition
A sequence is called a monotonic sequence if it is increasing, strictly increasing, decreasing, or strictly decreasing,

Examples
The following are all monotonic sequences:

1. The sequence $a_n=n$ (as it is strictly increasing).
2. The sequence $1,1,1,1,1, \dots $ (as it is increasing — and decreasing!).
3. The sequence $-1,-1,-2,-2,-3,-3,\dots $ (as it is decreasing).
4. The sequence $1/n$ (as it is strictly decreasing).

The following are not monotonic:

1. The sequence $a_n=(-1)^n$.
2. The sequence $a_n=n\cos(3n)$ since $a_1\approx-0.990$, $a_2\approx-1.920$, and $a_3\approx-2.733$.