Geometric Series

There’s an old story about the inventor of chess asking for a reward of rice: one grain on the first square of the board, two on the second, four on the third, doubling every square. It sounds modest until you add it up. The total is , about 18 quintillion grains, more rice than the world has ever grown. That sum is a geometric series, the total of a geometric sequence. Geometric series can explode like that, or, when the terms shrink, settle on a perfectly finite total even with infinitely many terms.

The Finite Geometric Series Formula

The sum of the first terms of a geometric sequence is

(If , every term is and the sum is simply .) In sigma notation the sum is .

Where the Formula Comes From

Write the sum, then multiply it by :

Subtract the second line from the first. Everything in the middle cancels, leaving . Factor and divide by to get the formula.

Example 1: A Growing Series

Find the sum of the first 8 terms of 3, 6, 12, …

Step 1 — the pieces: , , .

Step 2 — the formula:

Example 2: A Shrinking Series

Find .

Step 1 — the pieces: , , .

Step 2 — the formula:

The Chessboard

One grain on the first square, doubling across 64 squares, is , , :

Infinite Geometric Series

Now try adding infinitely many terms of a shrinking geometric sequence: . Each term is half of what’s left to reach 1, so the running total gets closer and closer to 1 without ever passing it.

When , the in the finite formula shrinks to as grows, which leaves the infinite geometric series formula:

For : , matching the picture.

When Does a Geometric Series Converge?

An infinite series converges if its running totals (its partial sums) settle on a single number, and diverges if they don’t.

So the question “which geometric series converges?” has a one-line answer: the one whose ratio is strictly between −1 and 1.

Example 3: An Infinite Sum

Find .

Step 1 — check convergence: , and .

Step 2 — the sum:

Repeating Decimals as Fractions

Every repeating decimal is an infinite geometric series in disguise.

Example 4. Write as a fraction.

Example 5. Write as a fraction. The repeating block has two digits, so and :

Worked Example: The Bouncing Ball’s Total Distance

A ball is dropped from 10 m, and each bounce reaches 80% of the previous height. Assuming it bounces forever, how far does it travel in total?

Step 1 — the first drop is 10 m. After that, each bounce goes up and comes back down, so each rebound height counts twice.

Step 2 — the rebound heights form an infinite geometric series: , with and :

Step 3 — the total distance:

Infinitely many bounces, but a finite 90 metres of travel.

Common Mistakes to Avoid

• Using the infinite formula when . doesn’t equal . It diverges.
• Using in the sum formula. The finite sum uses . The belongs to the term formula.
• Sign errors when . Both and are negative, so the negatives cancel. Some books write to avoid them; it’s the same formula.
• Using the wrong ratio for a repeating decimal. A two-digit repeating block means , not .
• Counting each bounce once. A bounce goes up and down.

Practice Problems

Work each one before opening the answer.

Problem 1. Find the sum of the first 6 terms of 2, 6, 18, …

Problem 2. Find .

Problem 3. Find .

Problem 4. Write as a fraction.

Problem 5. Does converge?

Problem 6. Evaluate .

Geometric series sit behind compound interest, mortgages and loan payments. The continuous compound interest lesson shows what happens when the compounding gets infinitely frequent. The last lesson in this chapter turns to a different kind of pattern: the coefficients you get when you expand .