Monotonicity of Sequences

Definition

Definition Increasing and Decreasing Sequences

Let $(u_n)$ be a sequence.

The sequence $(u_n)$ is increasing from index $n_0$ if for all $n \ge n_0$, $\boldsymbol{u_{n+1} \ge u_n}$.
The sequence $(u_n)$ is decreasing from index $n_0$ if for all $n \ge n_0$, $\boldsymbol{u_{n+1} \le u_n}$.
A sequence that is either increasing or decreasing is said to be monotonic.

Note

Just like with functions, if we replace the non-strict inequalities with strict ones, we speak of a strictly increasing or strictly decreasing sequence.
Some sequences are not monotonic, such as the sequence $(u_n)$ defined by $u_n = (-1)^n$, which alternates between $1$ and $-1$.

Studying the Variations of a Sequence

Method Studying the Sign of the Difference $u_{n+1}-u_n$

To study the monotonicity of a sequence $(u_n)$, we can determine the sign of the difference between two consecutive terms:

If for all $n\geq n_0$, $\textcolor{colorprop}{u_{n+1} - u_n \ge 0}$, the sequence is increasing from index $n_0$.
If for all $n\geq n_0$, $\textcolor{colorprop}{u_{n+1} - u_n \le 0}$, the sequence is decreasing from index $n_0$.

Example

Study the variations of the sequence $(u_n)$ defined by $u_n = 2n - 3$.

Answer

Let $n \in \mathbb{N}$.$$\begin{aligned}u_{n+1} - u_n &= [2(n+1) - 3] - [2n - 3] \\ &= 2n + 2 - 3 - 2n + 3 \\ &= 2\end{aligned}$$Since $2 > 0$, the sequence $(u_n)$ is strictly increasing.

Method Comparing the Ratio $\frac{u_{n+1}}{u_n}$ to 1

For a sequence with strictly positive terms ($u_n > 0$):

If for all $n\geq n_0$, $\textcolor{colorprop}{\frac{u_{n+1}}{u_n} \geq 1}$, the sequence is increasing from index $n_0$..
If for all $n\geq n_0$, $\textcolor{colorprop}{\frac{u_{n+1}}{u_n} \leq 1}$, the sequence is decreasing from index $n_0$..

Example

Study the variations of the sequence $(u_n)$ defined by $u_n = 2^n$.

Answer

Let $n \in \mathbb{N}$.
All terms are positive ($2^n > 0$).$$\begin{aligned} \frac{u_{n+1}}{u_n} &= \frac{2^{n+1}}{2^n}\\ &= 2^{n+1-n}\\ &= 2 \\ \end{aligned}$$Since $2 > 1$, the sequence $(u_n)$ is strictly increasing.