Lesson 19.1

Introduction to Sequences

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

Introduction

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

Past Knowledge

You can evaluate functions and understand domain notation.

Today's Goal

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

Future Success

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.