Lesson 19.1

Introduction to Sequences

Defining the n-th term and observing limits of sequences as n approaches infinity.

Introduction

A sequence is an ordered list of numbers following a specific pattern. Unlike a set, order matters—and we can ask what happens as we go further and further down the list. Does the sequence approach a limit, or does it grow without bound? This foundation leads directly to series and calculus.

Prerequisite Connection

You can evaluate functions and understand domain notation.

Today's Increment

We define sequences using explicit formulas and explore their long-term behavior.

Why This Matters

Sequences model growth patterns, population dynamics, and financial investments. Understanding their limits is essential for calculus.

Key Concepts

Definition: Sequence

A sequence is a function whose domain is the positive integers. We write for the -th term.

Explicit Formula

Gives directly in terms of :

Produces: 3, 5, 7, 9, ...

Recursive Formula

Defines each term using previous terms:

Same sequence: 3, 5, 7, 9, ...

Limit of a Sequence

If approaches a value as , we write:

If no such exists, the sequence diverges.

Worked Examples

Example 1: Finding Terms (Basic)

Find the first 4 terms of

Substitute n = 1, 2, 3, 4:

Sequence:

Example 2: Finding the Limit (Intermediate)

Find

Divide numerator and denominator by n:

As , :

The sequence converges to 1.

Example 3: Divergent Sequence (Advanced)

Determine if converges:

Write out the terms:

Observe the pattern:

The terms alternate between -1 and 1 forever.

The sequence DIVERGES

It oscillates and never settles on a single value.

Common Pitfalls

Confusing sequences with series

A sequence is a LIST of numbers. A series is the SUM of a sequence's terms. Don't mix them up!

Assuming bounded means convergent

is bounded between -1 and 1, but it doesn't converge because it oscillates.

Starting index confusion

Some sequences start at , others at . Always check the starting index.

Real-World Application

Population Growth Models

Biologists model population growth using sequences. If a population grows by a fixed percentage each year, gives the population after years. Understanding whether the sequence converges (stable population) or diverges (exponential growth/extinction) is critical for conservation planning.

If , population grows without bound. If , it decays toward 0.