Geometric Series: Bouncing Balls, Repeating Decimals, and Infinite Sums
October 29, 2026
A geometric series adds terms that each multiply the previous one by a constant ratio r. The sum of the first n terms is S = a × (1 − rⁿ) ÷ (1 − r), where a is the first term. If |r| < 1, the infinite series converges to S = a ÷ (1 − r).
Example: 1 + 1/2 + 1/4 + 1/8 + ... has a = 1 and r = 1/2, so the sum is 1 ÷ (1 − 0.5) = 2. After 10 terms, the sum is 2 × (1 − 1/1024) = 1.999, very close to 2.
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A ball is dropped from 10 feet, and each bounce reaches 60% of the previous height. The ball falls 10 feet. Then, on each bounce, it goes up and down the same distance: the first rebound is 6 feet up and 6 feet down, the next 3.6 up and 3.6 down, and so on. The up-and-down distances sum to 2 × (6 + 3.6 + 2.16 + ...) = 2 × 6 ÷ (1 − 0.6) = 2 × 15 = 30 feet. Total distance = 10 + 30 = 40 feet. The ball makes infinitely many bounces but travels a finite distance; with practical energy loss it stops. The time of the bounces also converges.
| Series | First term a | Ratio r | Sum |
|---|---|---|---|
| 1 + 1/2 + 1/4 + ... | 1 | 0.5 | 2 |
| 6 + 3.6 + 2.16 + ... | 6 | 0.6 | 15 |
| 0.9 + 0.09 + 0.009 + ... | 0.9 | 0.1 | 1 |
| 0.3 + 0.03 + 0.003 + ... | 0.3 | 0.1 | 1/3 |
Repeating Decimals
A repeating decimal is a geometric series: 0.333... = 0.3 + 0.03 + 0.003 + ... with a = 0.3 and r = 0.1, so the sum is 0.3 ÷ 0.9 = 1/3. Likewise, 0.999... = 0.9 ÷ (1 − 0.1) = 0.9 ÷ 0.9 = 1. That 0.999... equals exactly 1 surprises many, but it follows from the definition of the sum as the limit. Another example: 0.272727... = 0.27 ÷ (1 − 0.01) = 27/99 = 3/11.
Finite Sums in Finance
Saving $100 at the end of each month with 0.5% monthly interest: the future value is the sum of 100 × 1.005^k for k = 0 to 59 over five years, which is 100 × (1.005⁶⁰ − 1) ÷ 0.005 = 100 × 69.77 = $6,977. The loan payment formula and annuity values use the same geometric sum. A sequence that doubles each day (1, 2, 4, ...) for 30 days sums to 2³⁰ − 1 = 1,073,741,823.
Conditions and Pitfalls
The infinite sum formula only works if |r| < 1; with r = 1 or more, the series grows without bound. Be careful with the first term and the number of terms. Check the ratio by dividing consecutive terms. Alternating series with a negative ratio, such as 1 − 1/2 + 1/4 − ..., converge to a ÷ (1 − r) = 1 ÷ 1.5 = 2/3.
Frequently Asked Questions
What is the infinite geometric series sum?
a ÷ (1 − r) when |r| < 1.
How far does a ball dropped from 10 ft with 60% rebounds travel?
40 feet in total.
Why does 0.999... equal 1?
The series 0.9 + 0.09 + ... sums to 0.9 ÷ 0.9 = 1.
What is the finite sum formula?
S = a(1 − rⁿ) ÷ (1 − r).
Where is it used in finance?
Annuities, loan payments, and savings plans.