Calculus II · Unit 4A · lesson

Sequences as Functions on the Integers

Concept

Learning objectives

interpret sequence notation, generate terms from a formula, and graph sequences as discrete points.

Sequences as Functions on the Integers

Explanation

An infinite list still needs a precise rule

A sequence is an ordered list, but order alone is not enough for calculus. We treat a sequence as a function whose permitted inputs are positive integers. That viewpoint lets us use familiar function language - formula, domain, range, limit - while remembering that there are no sequence values between consecutive integers. A plotted sequence is therefore a collection of points rather than an unbroken curve.

The index carries meaning. In an application it may count payments, generations, bounces, or time steps. Changing the starting index can change notation without changing the underlying pattern, so every formula should be read together with its declared index set. The practical habit is simple: write several terms, label the starting index, and check that the formula actually matches the described list before attempting any limit.

Bridge

From a list to a mathematical object

Start with the list $1/2,2/3,3/4,\ldots$ . A pattern is visible, but calculus needs a rule that answers a sharper question: what is the term at position $n$ , and what happens when $n$ becomes arbitrarily large? Writing $a_n=n/(n+1)$ turns the pattern into a function rule and creates an object on which limit arguments can operate.

The discrete domain is not a minor technicality. We may draw a continuous guide curve to reveal shape, but the sequence has values only at its declared integer inputs. Throughout this unit, continuous functions will often help us reason about sequences, yet every conclusion must return to the actual integer-indexed terms.

**The index selects one term at a time**
A sequence is a function whose allowed inputs are integers. Each input selects one ordered term.
The index selects one term at a time. Index-to-term machine plus discrete point graph.

How to read the visual

The order and the integer domain are part of the definition, not decoration.

**A sequence is discrete, not a continuous curve**
The points of a sequence occur only at positive integer inputs. A faint guide curve may reveal a pattern, but the curve is not part of the sequence.
A sequence is discrete, not a continuous curve. Index-to-term machine plus discrete point graph.

How to read the visual

The marked values rise toward the dashed level $1$ . The horizontal spacing is discrete: there is no sequence term at $n=2.4$ .

Concept

Core notation

A sequence may be written $\{a_n\}_{n=1}^{\infty}$ , $a_1,a_2,a_3,\ldots$ , or as a rule such as

a_n=\frac{n}{n+1}.

The index $n$ is an integer input; $a_n$ is the corresponding output.

Guided walkthrough

Generate terms and identify a pattern

For $a_n=(-1)^n\frac{n}{n+1}$ , the first four terms are

a_1=-\frac12,\quad a_2=\frac23,\quad a_3=-\frac34,\quad a_4=\frac45.

The magnitudes rise toward $1$ , while the signs alternate. Writing terms before describing behavior prevents the common mistake of ignoring the factor $(-1)^n$ .

Worked example

One sequence can combine trend and oscillation

Consider $b_n=2+(-1)^n/n$ . Its first terms are

1,\quad \frac52,\quad \frac53,\quad \frac94,\quad \frac95,\ldots

The factor $(-1)^n$ makes the terms alternate around $2$ , while $1/n$ makes the oscillation shrink. A complete description must mention both behaviors. Saying only that the sequence "goes up and down" misses the limiting structure.

Common mistake

A guide curve is not the sequence

Values such as $a_{2.5}$ do not exist for a sequence indexed by positive integers. A curve through the points may suggest behavior, but it does not add new terms.

Interactive checku4a-sequences_as_functions-01

For $a_n=\frac{n}{n+1}$ , find $a_5$ .

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Show hint

Substitute $n=5$ into the rule.

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Exercise

List the first five terms of $a_n=2-3/n$ .

Exercise

Graph the first eight terms of $a_n=(-1)^n/n$ .

Exercise

Explain why a line drawn through sequence points is only a visual guide.

Exercise

A bacteria count doubles each generation from an initial count of 50. Write a sequence formula.

After the explanation

Use the section idea

Reading lens

Track the integer domain, late-term behavior, monotonicity, bounds, and any recurrence before asserting a limit.

Mental model

A sequence converges when every sufficiently late term remains arbitrarily close to one finite target.

Decision

Use algebraic limits when a formula is explicit; use bounds and monotonicity when a recurrence hides the formula.

Common trap

Reading a finite plot as proof or solving a recurrence's fixed-point equation before proving convergence.

Check yourself

Can you justify both the candidate limit and why the terms must approach it?

Source & rights

Original instruction with traceable references.

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