A geometric series converges when the absolute value of its common ratio is less than 1, and diverges when the absolute value is 1 or greater. That single number, the common ratio (r), is the only thing you need to check. Once you know how to find it and evaluate it, you can classify any geometric series in seconds.
The Common Ratio Decides Everything
Every geometric series has a common ratio, usually called r. It’s the factor you multiply by to get from one term to the next. The convergence rule is straightforward:
- |r| < 1: The series converges (adds up to a finite number)
- |r| ≥ 1: The series diverges (grows without bound or oscillates indefinitely)
That’s the entire test. If r is 0.5, the series converges. If r is −0.75, it converges (because the absolute value is 0.75). If r is 1, −1, 2, or −3, the series diverges. The sign of r doesn’t matter for convergence. Only the size of r matters.
Why does this work intuitively? When |r| is less than 1, each successive term gets smaller. The terms shrink so rapidly that their infinite sum still lands on a finite value. When |r| equals or exceeds 1, the terms either stay the same size or grow, so the running total never settles down.
How to Find the Common Ratio
If you’re given a list of terms, divide any term by the one before it. That quotient is r. For the sequence 6, 3, 1.5, 0.75, dividing 3 by 6 gives 0.5. Check another pair: 1.5 ÷ 3 = 0.5. The common ratio is 0.5, and since |0.5| < 1, the series converges.
If you’re given the series in summation notation, like ∑ a·rⁿ, the ratio is written directly in the expression. Look for the base that’s being raised to a power involving n or k. That base is your r. For example, in ∑ 5·(1/3)ⁿ, the common ratio is 1/3.
One common mistake: confusing the first term with the ratio. The first term (a) is the coefficient out front or the value you get when you plug in your starting index. It tells you the scale of the series but has no effect on whether the series converges or diverges. A series with a first term of 1,000,000 and r = 0.1 still converges. A series with a first term of 0.001 and r = 2 still diverges.
Finding the Sum When It Converges
Once you’ve confirmed |r| < 1, you can calculate the exact value the series adds up to. The formula is:
S = a / (1 − r)
Here, a is the first term of the series and r is the common ratio. For the series 6 + 3 + 1.5 + 0.75 + …, the first term is 6 and r is 0.5. Plugging in: S = 6 / (1 − 0.5) = 6 / 0.5 = 12. So this infinite series adds up to exactly 12.
For the series 10 − 5 + 2.5 − 1.25 + …, the common ratio is −0.5 (each term is the previous term multiplied by −0.5). Since |−0.5| = 0.5 < 1, it converges. The sum is 10 / (1 − (−0.5)) = 10 / 1.5 ≈ 6.667.
This formula only works for infinite geometric series that converge. If |r| ≥ 1, the sum doesn’t exist as a finite number, and plugging into the formula would give a meaningless result.
Watch the Starting Index
A subtle point that trips people up in textbook problems: the formula S = a / (1 − r) assumes “a” is the actual first term of the series. Some problems write the series starting at n = 0, like ∑ from n=0 to ∞ of arⁿ, where the first term is a·r⁰ = a. Others start at n = 1, like ∑ from n=1 to ∞ of arⁿ⁻¹, where the first term is still a. Both forms give the same result.
But if you see ∑ from n=1 to ∞ of arⁿ (note: the exponent is n, not n−1), the first term is a·r, not a. In that case, you’d use the first term as a·r in the formula, or equivalently compute ar / (1 − r). The safest approach: plug in your starting index value to find the actual first term, then use that as “a” in S = a / (1 − r).
Worked Examples
Example 1: A Convergent Series
Determine whether 100 + 80 + 64 + 51.2 + … converges or diverges.
Find r: 80 ÷ 100 = 0.8. Verify: 64 ÷ 80 = 0.8. Since |0.8| < 1, the series converges. Its sum is 100 / (1 − 0.8) = 100 / 0.2 = 500.
Example 2: A Divergent Series
Determine whether 3 + 4.5 + 6.75 + 10.125 + … converges or diverges.
Find r: 4.5 ÷ 3 = 1.5. Since |1.5| ≥ 1, the series diverges. The terms keep growing, so the sum increases without limit. Don’t bother with the sum formula.
Example 3: Negative Ratio
Determine whether 12 − 4 + 4/3 − 4/9 + … converges or diverges.
Find r: −4 ÷ 12 = −1/3. Since |−1/3| = 1/3 < 1, the series converges. The alternating signs mean the partial sums bounce above and below the final value, but they still zero in on it. The sum is 12 / (1 − (−1/3)) = 12 / (4/3) = 9.
Example 4: Ratio of Exactly 1 or −1
The series 5 + 5 + 5 + 5 + … has r = 1. Every term is the same, and the sum grows forever. It diverges. The series 5 − 5 + 5 − 5 + … has r = −1. The partial sums alternate between 5 and 0 endlessly, never settling on a value. It also diverges.
Repeating Decimals as Geometric Series
One of the most satisfying applications: any repeating decimal is a convergent geometric series in disguise. Take 0.333… This equals 3/10 + 3/100 + 3/1000 + …, which is a geometric series with first term 3/10 and r = 1/10. Since |1/10| < 1, it converges. The sum is (3/10) / (1 − 1/10) = (3/10) / (9/10) = 3/9 = 1/3. That’s exactly the fraction you’d expect.
The same logic converts 0.272727… into a fraction. The series is 27/100 + 27/10000 + …, with a = 27/100 and r = 1/100. The sum is (27/100) / (1 − 1/100) = (27/100) / (99/100) = 27/99 = 3/11.
Quick Reference: The Decision Process
When you encounter a geometric series, run through these steps in order:
- Confirm it’s geometric: Each term divided by the previous term gives the same value.
- Find r: Divide any term by the one before it, or read it from the summation notation.
- Check |r|: If less than 1, it converges. If 1 or greater, it diverges.
- Calculate the sum (if convergent): Identify the first term, then use S = a / (1 − r).
The common ratio is the only variable that controls convergence. The first term, no matter how large or small, affects the value of the sum but never whether the sum exists.