Lesson 19.7

Infinite Geometric Series

Determining when a series converges (|r| < 1) and calculating the sum S = a₁/(1-r).

Introduction

What happens when we add infinitely many terms? Surprisingly, if the terms shrink fast enough, the sum approaches a finite limit. An infinite geometric series converges when, and its sum has a beautifully simple formula.

Prerequisite Connection

You can find finite geometric sums using .

Today's Increment

We take to derive when .

Why This Matters

Infinite series model repeating decimals, present value of perpetuities, and are foundational to calculus.

Key Concepts

Convergence Condition

An infinite geometric series converges if and only if:

That is,

Sum Formula (when |r| < 1)

This is the limit of as

Why It Works

When , as :

Divergence: |r| ≥ 1

If , the series diverges—the sum grows without bound or oscillates.

Worked Examples

Example 1: Simple Convergent Series (Basic)

Find the sum:

Since , it converges.

Example 2: Repeating Decimal (Intermediate)

Express as a fraction.

Write as series:

Example 3: Convergence Check (Advanced)

Does converge? If so, find the sum.

✓ Converges!

Common Pitfalls

Using formula when |r| ≥ 1

The formula only works when . Always check convergence first!

Sign errors with negative r

, not

Forgetting absolute value

converges because

Real-World Application

Perpetuities in Finance

A perpetuity pays a fixed amount forever. If you receive $100/year with 5% discount rate:

This is the infinite geometric series formula in disguise!