A geometric series diverges whenever the absolute value of the common ratio is greater than or equal to 1. In mathematical notation, that means the series diverges when |r| ≥ 1. If the common ratio r falls strictly between -1 and 1, the series converges to a finite sum. Anything outside that window, and the series has no finite total.
The Common Ratio Rule
Every geometric series has the form a + ar + ar² + ar³ + … , where “a” is the first term and “r” is the common ratio (the number you multiply by each time to get the next term). The entire question of convergence or divergence comes down to the size of r.
When |r| < 1, each successive term gets smaller. The partial sums gradually approach a finite limit, and the series converges to a/(1 - r). When |r| ≥ 1, the terms either stay the same size or grow, so the running total never settles on a single value. That's divergence.
There are three distinct ways a geometric series can diverge, depending on whether r is greater than 1, equal to 1, less than or equal to -1, or exactly -1. Each case looks a little different.
When r Is Greater Than 1
If r > 1, every term is larger than the one before it. Take the series 3 + 6 + 12 + 24 + …, where r = 2. The partial sums (3, 9, 21, 45, 93, …) grow without bound, racing toward infinity. No matter how far out you go, the total keeps climbing faster and faster. This is the most intuitive type of divergence: the numbers just get bigger.
When r Equals 1
If r = 1, every term is identical to the first. The series becomes a + a + a + a + …, and the partial sum after n terms is simply n × a. That sum grows forever (assuming a isn’t zero), so the series diverges. There’s also a formula problem here: the standard convergence formula a/(1 – r) would require dividing by (1 – 1), which is zero. The formula is literally undefined, which is a signal that convergence doesn’t apply.
When r Equals -1
This case is subtler. When r = -1, the terms alternate in sign: a, -a, a, -a, a, -a, and so on. The partial sums bounce back and forth between two values. After an odd number of terms, the sum equals a. After an even number of terms, it equals 0. The sum never settles on one number. It just keeps oscillating forever.
This is sometimes called oscillatory divergence. The series doesn’t blow up to infinity, but it still diverges because there’s no single value the partial sums approach.
When r Is Less Than -1
If r < -1, you get the worst of both worlds: the terms alternate in sign and grow in size. Consider r = -2 with a first term of 1. The series is 1, -2, 4, -8, 16, -32, ... The partial sums swing wildly in both directions, getting farther from zero with each step. The series diverges because the partial sums have no limit.
Why the Sum Formula Breaks Down
The finite partial sum of the first n + 1 terms of a geometric series is (1 – r^(n+1)) / (1 – r). When |r| < 1, the r^(n+1) piece shrinks toward zero as n grows, leaving a clean result: 1/(1 - r), multiplied by the first term. That's where the familiar convergence formula comes from.
When |r| ≥ 1, the r^(n+1) term doesn’t shrink. If r > 1, it explodes toward infinity. If r = -1, it flips between +1 and -1 forever. If r < -1, it alternates sign while growing in size. In every one of these cases, the partial sum formula never approaches a fixed number, so no finite "sum to infinity" exists.
Quick Reference
- |r| < 1: Converges. Sum = a / (1 – r).
- r = 1: Diverges. Partial sums grow without bound.
- r = -1: Diverges. Partial sums oscillate between two values.
- r > 1: Diverges. Partial sums increase toward infinity.
- r < -1: Diverges. Partial sums swing between increasingly large positive and negative values.
Spotting Divergence in Practice
To check whether a geometric series diverges, identify the common ratio by dividing any term by the one before it. If the result has an absolute value of 1 or more, the series diverges. You don’t need to compute partial sums or apply any convergence test beyond this single check.
One common mistake is applying the sum formula a/(1 – r) without first verifying that |r| < 1. If you plug in r = 2, the formula spits out a negative number, which makes no sense for a series of positive, growing terms. The formula is only valid inside the convergence interval. Outside it, the result is meaningless.
For power series (where the common ratio contains a variable, like x²/3), you find the interval of convergence by setting the absolute value of that expression less than 1 and solving for x. Any x value outside that interval produces a divergent geometric series.