Monotonic Sequences - Mathematics.lu

In the previous page we introduced sequences formally and discussed some different types of sequences: defined explicitly, recursively, arithmetic and geometric sequences.

In this page we discuss sequences whose behaviour is monotonic. That is to say, they are always increasing or decreasing.

Table of Contents

Basic Inequalities

In previous years, there is not much of a discussion about inequalities when suddenly in S6 parts of the programme require you to understand them on an intuitive level. This section is meant to explain inequalities at a basic level with the hopes of understanding them more intuitively as the exercises/examples progress.

In S5 you discussed quadratic models and you have seen that the graph of a function $f(x)=ax^2+bx+c$ is a parabola. It can intersect the $x$ -axis at most twice. Once you draw the graph, it is easy to see where the parabola is above the $x$ -axis (i.e.: $f(x)>0$ ) or below the $x$ -axis (i.e.: $f(x)<0$ ). We draw our conclusions accordingly. A way to simplify this is by drawing a sign diagram of the function.

A sign diagram of a function $f(x)$ aims to represent the function minimally, by marking where the function is positive, negative, or zero. Let us see an example of how we use this to solve a quadratic inequality:

Example: Solve the quadratic inequality:

$x^2 + 4x + 3 > 0$

In summary, we have solved this inequality by understanding where the function is zero, and checking the regions before, after, and in between these $x$ -intercepts. This is generally a good strategy to solve simple inequalities. You should also include points of discontinuity in your sign diagram, but this is something that might become more intuitive in the next unit. Just think of this last remark the following way: The function $\frac{1}{x}$ is never zero, but it is very easy to understand its sign diagram. We include 0 in its sign diagram because the function is not defined at 0, so this is a point of discontinuity.

You should consider these as they are points where your function could “disappear” and “reappear” on the other side of the $x$ -axis. In short, a function cannot change its sign without going through the $x$ -axis or having an asymptote/point of discontinuity somewhere. This remark is only meant for completeness, most cases can be tackled with a relatively straightforward level of analysis.

A few simple rules about inequalities:

We treat/solve them as we do equations. We can make the same change on both sides to end up with an equivalent expression. E.g.: $2x>6\iff x>3$ . Here we divided both sides by 2 and both inequalities are equivalent.
If, at any point, we multiply or divide by a negative number, we must flip the inequality sign. E.g.:
$-2x>8\iff x<-4$
We cannot divide by something that could potentially be zero. In that case, we should separate into two cases and study them separately (once assuming the expression is 0, once assuming it is not).

Let us now see an example of an inequality involving a rational function (quotient of two polynomials) if only to try and see how inequalities are not very different from equations if we simply use a bit of logic.

Example: Solve the rational inequality:

$\frac{x + 2}{x - 3} > 0$

We could have also asked for $\frac{x+2}{x-3}\geq0$ , in which case we should include the value $x=-2$ in our solution. The value $x=3$ will not be included because the denominator then would be zero, and we cannot divide an expression by 0.

Monotonically Increasing/Decreasing Sequences

We begin with formal definitions:

Definition (Increasing and Decreasing Sequences): A sequence $\{u_n\}_{n=1}^{\infty}$ is classified by how its consecutive terms change as the position index $n$ grows larger.

1. Increasing Sequence

A sequence is increasing if each term is greater than or equal to the preceding term for all positive integers:

$u_{n+1} \ge u_n \quad \text{for all } n \ge 1$

Note: If the inequality is strict ( $u_{n+1} > u_n$ ), the sequence is strictly increasing.

2. Decreasing Sequence

A sequence is decreasing if each term is less than or equal to the preceding term for all positive integers:

$u_{n+1} \le u_n \quad \text{for all } n \ge 1$

Note: If the inequality is strict ( $u_{n+1} < u_n$ ), the sequence is strictly decreasing.

Sequences that are either completely increasing or completely decreasing are referred to as monotonic sequences.

Alternatively, we can compare:

$\frac{u_{n+1}}{u_n}>1$ in which case the sequence is increasing,
$\frac{u_{n+1}}{u_n}<1$ in which case it is decreasing,
$u_{n+1}-u_n>0$ , i.e.: increasing,
$u_{n+1}-u_n<0$ , i.e.: decreasing.

Example: Determine whether the sequence defined explicitly by $u_n = \frac{n}{n+1}$ is increasing or decreasing for all $n \ge 1$ .

Constant Sequences

A sequence $\{u_n\}_{n=1}^\infty$ is constant if $u_1=u_2=...=u_n=k$ for any $n\in\mathbb{N}$ with $k$ being some constant, real number.

This can also be shown as $\frac{u_{n+1}}{u_n}=k$ , but in practice it should be obvious when a sequence is constant.

Graphing Sequences

We have seen that some sequences are defined explicitly. For example $u_n=\frac{1}{n}$ . Here we have $u_1=\frac{1}{1},u_2=\frac{1}{2},...$ .

What we are doing to find a term is replacing $n$ in the formula for $u_n$ as we do with functions. When we want to find $f(2)$ we replace $x=2$ in its equation. The big difference being that in a sequence $u_n$ , the term $n$ can only be a non-negative integer. Either way, just as we replace values of $n$ we can represent this sequence graphically.

Example: Generate the first 4 terms of the sequence defined by $u_n = \frac{1}{n}$ for $n \ge 1$ , and plot its terms to observe its long-term behavior.

Substituting index values $n = 1, 2, 3, 4$ into the explicit general formula yields the first 4 terms:

• $u_1 = \frac{1}{1} = 1$
• $u_2 = \frac{1}{2} = 0.5$
• $u_3 = \frac{1}{3} \approx 0.333$
• $u_4 = \frac{1}{4} = 0.25$

Rendered by QuickLaTeX.com

The terms of the sequence are $\{1, 0.5, \frac{1}{3}, 0.25\}$ . As shown on the plotted grid points, each additional discrete step drops closer to the horizontal axis.

Can we conclude from the graph that the sequence $u_n=\frac{1}{n}$ ? No! Seeing a few terms is not enough to conclude such behaviour. Instead, it is infinitely better to show it by definition.

$\frac{u_{n+1}}{u_n}=\frac{\frac{1}{n+1}}{\frac{1}{n}}=\frac{1}{n+1}\cdot\frac{n}{1}=\frac{n}{n+1}<1$

We can conclude that $\frac{n}{n+1}<1$ because we see that the denominator is greater than the numerator. I.e.: $n<n+1$ .

Exercises

For each of the following sequences defined for $n \ge 1$ , determine whether the sequence is increasing or decreasing.

$u_n = \frac{1}{n}$

$u_n = \frac{n}{n + 2}$

$u_n = \frac{2n - 1}{n}$

$u_n = \frac{3}{n + 1}$

$u_n = \frac{n^2}{n^2 + 1}$

Analyze the mathematical behavior of the sequences below by comparing consecutive terms or studying algebraic trends for all positive integers $n \ge 1$ .

Prove whether the sequence explicitly defined below is increasing, decreasing, or non-monotonic by evaluating the sign of the difference $u_{n+1} - u_n$ :

$u_n = \frac{1}{2n + 3}$

Show that the sequence defined below is strictly decreasing, and determine its lower limit value as $n$ approaches infinity ( $\infty$ ):

$u_n = 5 + \left(\frac{1}{3}\right)^n$

Analyze the behavior of the sequence defined explicitly below for $n \ge 1$ . No calculators are allowed.

Generate the first 4 terms of the sequence, determine what type of sequence it is, and classify its monotonicity:

$u_n = (-1)^{2n}$

For each of the following three sequences defined for $n \ge 1$ , calculate the first 4 terms. Then, sketch their discrete coordinate points on an axis system containing only the first quadrant, showing enough terms to illustrate their long-term behavior.

$u_n = \frac{n}{n + 1}$

$u_n = 2 - \frac{1}{n}$

$u_n = 2n$

Consider the sequence defined explicitly for $n \ge 1$ by the quadratic formula:

$u_n = (n - 5)^2$

Complete the following tasks to analyze how its behavior changes across different index ranges.

Calculate the numeric values of the first 7 terms ( $u_1$ through $u_7$ ) of the sequence.

Plot these 7 terms as discrete coordinate points $(n, u_n)$ in the first quadrant of an axis system.

Using your calculated terms and plot, conclude why we cannot claim that a sequence is increasing or decreasing just by looking at parts of its graph.

Identify the specific point (index value $n$ ) from which the sequence turns and becomes strictly increasing for all subsequent terms. Prove this trend by examining the inequality $u_{n+1} > u_n$ .

Analyze the behavior of the alternating sequence defined explicitly below for $n \ge 1$ .

Generate the first 4 terms of the sequence, and use them to determine if the sequence is increasing, decreasing, or neither:

$u_n = (-1)^n$