Monotone and Bounded Sequences

students, in this lesson you will learn two of the most important ideas for understanding sequences in Calculus 2: monotone sequences and bounded sequences 📈. These ideas help us predict whether a sequence keeps moving in one direction and whether its terms stay trapped between two fixed numbers. That matters because sequences often appear in compound interest, repeating processes, computer algorithms, and series.

Lesson objectives:

Explain what it means for a sequence to be monotone or bounded.
Use examples to decide whether a sequence is monotone, bounded, both, or neither.
Apply the Monotone Convergence Theorem to understand when a sequence converges.
Connect these ideas to the larger study of sequences and series.

By the end, students, you should be able to look at a sequence and describe its behavior using precise mathematical language.

What Is a Sequence?

A sequence is an ordered list of numbers, usually written as $\{a_n\}$ or $\{a_1, a_2, a_3, \dots\}$. The index $n$ tells us the position of each term. For example, the sequence

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

gives the terms $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$.

In Calculus 2, we often study whether the terms get closer to a number as $n$ gets large. But before we can do that, it helps to understand two key patterns:

Monotone behavior: does the sequence go up or go down?
Bounded behavior: does the sequence stay within fixed limits?

These ideas are simple, but they are powerful tools for proving convergence.

Monotone Sequences: Always Going One Direction

A sequence is monotone increasing if its terms never decrease. In symbols, $\{a_n\}$ is monotone increasing if

$$a_{n+1} \ge a_n \quad \text{for all } n.$$

If each term is strictly larger than the one before it, then the sequence is strictly increasing, meaning

$$a_{n+1} > a_n \quad \text{for all } n.$$

A sequence is monotone decreasing if its terms never increase. In symbols,

$$a_{n+1} \le a_n \quad \text{for all } n.$$

If each term is strictly smaller than the one before it, then it is strictly decreasing, meaning

$$a_{n+1} < a_n \quad \text{for all } n.$$

Example 1: A monotone increasing sequence

Consider

$$a_n = 3 - \frac{1}{n}.$$

The first few terms are $2, 2.5, 2.666\dots, 2.75, \dots$. Each term is larger than the previous one. In fact,

$$a_{n+1} - a_n = \left(3 - \frac{1}{n+1}\right) - \left(3 - \frac{1}{n}\right) = \frac{1}{n} - \frac{1}{n+1} > 0.$$

So the sequence is strictly increasing.

Example 2: A monotone decreasing sequence

Consider

$$b_n = \frac{5}{n}.$$

The terms are $5, \frac{5}{2}, \frac{5}{3}, \frac{5}{4}, \dots$. Since

$$b_{n+1} - b_n = \frac{5}{n+1} - \frac{5}{n} < 0,$$

the sequence is strictly decreasing.

Why monotonicity matters

If a sequence moves in only one direction, that makes it easier to analyze. For example, if a sequence is increasing but never gets too large, then it may approach a limit. If it is decreasing but never gets too small, it may also converge. This is the heart of the Monotone Convergence Theorem.

Bounded Sequences: Trapped Between Two Numbers

A sequence is bounded above if there is a number $M$ such that

$$a_n \le M \quad \text{for all } n.$$

It is bounded below if there is a number $m$ such that

$$a_n \ge m \quad \text{for all } n.$$

If a sequence is both bounded above and bounded below, it is simply called bounded. That means all its terms stay inside some interval:

$$m \le a_n \le M \quad \text{for all } n.$$

Example 3: A bounded sequence

Take

$$c_n = \frac{1}{n}.$$

Every term is positive, so

$$c_n \ge 0,$$

and every term is at most $1$, so

$$c_n \le 1.$$

Therefore, the sequence is bounded. It is trapped between $0$ and $1$.

Example 4: An unbounded sequence

Consider

$$d_n = n.$$

The terms are $1, 2, 3, 4, \dots$. This sequence keeps growing without limit, so there is no number $M$ such that

$$n \le M \quad \text{for all } n.$$

Thus, it is not bounded above, even though it is bounded below by $1$.

Bounded does not mean convergent

A bounded sequence does not always converge. For example,

$$e_n = (-1)^n$$

alternates between $-1$ and $1$. It is bounded because

$$-1 \le (-1)^n \le 1,$$

but it does not approach a single number. So boundedness alone is not enough to guarantee convergence.

The Monotone Convergence Theorem

One of the most important results in this topic is the Monotone Convergence Theorem.

If a sequence is monotone and bounded, then it converges.

More precisely:

If $\{a_n\}$ is monotone increasing and bounded above, then $\{a_n\}$ converges.
If $\{a_n\}$ is monotone decreasing and bounded below, then $\{a_n\}$ converges.

This theorem is a major reason these two ideas are studied together. Monotonicity tells us the sequence moves in one direction, and boundedness prevents it from escaping forever.

Example 5: Using the theorem

Let

$$a_n = 2 - \frac{1}{n}.$$

We already saw that this sequence is increasing. Also,

$$a_n < 2$$

for all $n$, so it is bounded above by $2$. Therefore, by the Monotone Convergence Theorem, the sequence converges.

In fact, we can find the limit:

$$\lim_{n \to \infty}\left(2 - \frac{1}{n}\right) = 2.$$

The theorem told us convergence would happen; the limit calculation tells us where it goes.

Example 6: A decreasing bounded sequence

Let

$$b_n = \frac{n}{n+1}.$$

The first few terms are $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \dots$. This sequence is increasing, not decreasing, and it is bounded above by $1$ because

$$\frac{n}{n+1} < 1.$$

So it converges by the theorem. In fact,

$$\lim_{n \to \infty} \frac{n}{n+1} = 1.$$

students, notice how the theorem does not require you to know the limit first. It lets you prove convergence from structure.

How to Check Monotone and Bounded Behavior

When given a sequence, there are several common strategies.

1. Compare consecutive terms

Look at

$$a_{n+1} - a_n.$$

If $a_{n+1} - a_n \ge 0$, the sequence is increasing.
If $a_{n+1} - a_n \le 0$, the sequence is decreasing.

This is one of the most direct methods.

2. Use algebra or inequalities

Sometimes it is easier to rewrite the sequence or compare it to a known number.

For example, if

$$a_n = \frac{n}{n+1},$$

then since $n < n+1$, dividing by the positive number $n+1$ gives

$$\frac{n}{n+1} < 1.$$

That proves an upper bound.

3. Think about the graph or pattern

A graph of the terms can give a visual clue. If the points rise steadily, the sequence may be increasing. If they level off toward a value, the sequence may be bounded and convergent.

4. Watch for oscillation

Sequences like

$$(-1)^n, \quad \sin(n), \quad \text{or} \quad \frac{(-1)^n}{n}$$

may bounce up and down instead of moving in one direction. Such sequences are often not monotone, even if they are bounded.

Connection to Sequences and Series

Monotone and bounded sequences are not just a side topic. They are a foundation for later work with series.

A series is formed by adding the terms of a sequence:

$$\sum_{n=1}^{\infty} a_n.$$

To understand whether a series converges, we often study the behavior of its partial sums:

$$S_n = \sum_{k=1}^{n} a_k.$$

The partial sums themselves form a sequence. So when you study sequences, you are also preparing to study series.

For example, the geometric series

$$\sum_{n=0}^{\infty} ar^n$$

has partial sums that may be monotone and bounded when $0 < r < 1$ and $a > 0$. In that case, the partial sums increase but stay below a fixed value, so they converge.

This is one reason monotone and bounded sequences matter so much: they help explain why many infinite processes settle down to a finite answer.

Conclusion

students, monotone and bounded sequences are core ideas in Calculus 2 because they help us understand long-term behavior. A monotone sequence moves in one direction, either increasing or decreasing. A bounded sequence stays within fixed limits. When a sequence has both properties, the Monotone Convergence Theorem guarantees that it converges.

These ideas are useful far beyond this lesson. They appear in limits of sequences, partial sums of series, numerical methods, and many real-world models. If you can recognize monotonicity and boundedness, you have a strong tool for deciding whether a sequence will settle to a limit 📘.

Study Notes

A sequence is an ordered list of numbers written as $\{a_n\}$.
Monotone increasing means $a_{n+1} \ge a_n$ for all $n$.
Strictly increasing means $a_{n+1} > a_n$ for all $n$.
Monotone decreasing means $a_{n+1} \le a_n$ for all $n$.
Strictly decreasing means $a_{n+1} < a_n$ for all $n$.
A sequence is bounded above if $a_n \le M$ for some number $M$ and all $n$.
A sequence is bounded below if $a_n \ge m$ for some number $m$ and all $n$.
A sequence is bounded if it is both bounded above and bounded below.
The sequence $\{(-1)^n\}$ is bounded but not monotone.
The sequence $\{n\}$ is monotone increasing but not bounded above.
The sequence $\left\{3 - \frac{1}{n}\right\}$ is monotone increasing and bounded above, so it converges.
The Monotone Convergence Theorem says: monotone + bounded = convergent.
Partial sums of series form sequences, so these ideas connect directly to series.