Infinite Geometric Series
Derive the sum formula S = a₁/(1−r) for infinite geometric series when |r| < 1. Understand why the series diverges for |r| ≥ 1. Apply to repeating decimals and Zeno's paradox.
A lesson of Algebra 2, in Sequences, Series & Probability, about 10 minutes.
By the end of this lesson, you will be able to determine if an infinite geometric series converges or diverges, compute the sum of a convergent series, and apply this to repeating decimals and real-world models.
The Big Idea
An infinite series — adding up infinitely many terms — can still have a finite sum if the terms shrink fast enough. For geometric series with $|r| < 1$, each term gets smaller and smaller, and the sum approaches a finite limit. When $|r| \geq 1$, the terms don't shrink and the series grows without bound.
Convergence Criterion
An infinite geometric series converges if and only if $|r| < 1$.
$$S_\infty = \frac{a_1}{1-r} \quad \text{when } |r| < 1$$
Diverges (no finite sum) when $|r| \geq 1$.
Derivation: $S_n = a_1\dfrac{1-r^n}{1-r}$. As $n \to \infty$, $r^n \to 0$ when $|r| < 1$:
$S_\infty = \dfrac{a_1}{1-r}$
Example 1 — Basic Infinite Sum
$\sum_{n=1}^{\infty} \frac{1}{2^n} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$
$a_1 = \frac{1}{2}$, $r = \frac{1}{2}$
$S_\infty = \dfrac{1/2}{1 - 1/2} = \dfrac{1/2}{1/2} = 1$ ✓
(The sum of all powers of $1/2$ equals 1 — each step covers half the remaining distance to 1.)
Example 2 — General Case
$6 + 2 + \frac{2}{3} + \frac{2}{9} + \cdots$
$a_1 = 6$, $r = \frac{1}{3}$
$S_\infty = \dfrac{6}{1 - 1/3} = \dfrac{6}{2/3} = 9$
Example 3 — Divergent Series
$3 + 6 + 12 + 24 + \cdots$: $r = 2 > 1$ → diverges. No finite sum.
$5 - 5 + 5 - 5 + \cdots$: $r = -1$, $|r| = 1$ → diverges. The partial sums oscillate between 0 and 5.
Repeating Decimals as Infinite Geometric Series
Every repeating decimal is an infinite geometric series.
$0.\overline{3} = 0.333\ldots$
$= \frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \cdots = \frac{3/10}{1-1/10} = \frac{3/10}{9/10} = \frac{3}{9} = \frac{1}{3}$ ✓
$0.\overline{142857} = \frac{1}{7}$ — the same technique works for any repeating block.
Example 4 — Application: Bouncing Ball
A ball is dropped from 10 feet and bounces to 60% of its previous height each time. Find the total distance traveled.