How To Tell If A Geometric Series Converges Or Diverges?

A geometric series converges when the absolute value of its common ratio is less than 1, and diverges when that absolute value is 1 or greater. The test is that simple. You find the ratio by dividing any term by the term before it, check whether it falls between -1 and 1, and the answer follows immediately. This article walks through the ratio test, shows worked examples, and covers the cases that trip people up.

What Makes a Series Geometric?

A geometric series is a sum where each term is the previous term multiplied by the same fixed number. That number is called the common ratio, usually written as r.

Take the series 1 + 1/2 + 1/4 + 1/8 + … Each term is half the one before it, so r = 1/2. Or 3 + 6 + 12 + 24 + …, where each term doubles, so r = 2. The pattern is always multiplication by a constant.

In symbols, a geometric series looks like a + ar + ar² + ar³ + …, where a is the first term and r is the ratio. The nth term is ar^(n-1).

Not every series is geometric. The series 1 + 1/2 + 1/3 + 1/4 + … is not, because the ratio between consecutive terms changes. It shrinks toward 1 but never settles on a single value. That distinction matters, because the convergence test below only applies to geometric series.

How To Tell If A Geometric Series Converges Or Diverges

The rule rests entirely on the size of r, ignoring its sign. Compare |r|, the absolute value of the common ratio, to 1.

• If |r| < 1, the series converges. The terms shrink toward zero fast enough that the running total settles on a finite number.
• If |r| ≥ 1, the series diverges. The terms either stay the same size, grow, or oscillate without settling, and the sum does not approach any finite value.

That is the whole test. Find r, take its absolute value, and compare to 1.

The reason it works is straightforward. When |r| < 1, each term is a fixed fraction of the one before it, so the terms collapse toward zero quickly. The total added up forever stays bounded. When |r| ≥ 1, the terms do not shrink. They hold steady or grow, so the sum keeps climbing or bounces back and forth without landing anywhere.

How Do You Find the Common Ratio?

Divide any term by the term immediately before it. The result is r.

For the series 2 + 6 + 18 + 54 + …, divide 6 by 2 to get 3. Check with another pair: 18 ÷ 6 = 3. The ratio is consistent, so r = 3. Since |3| ≥ 1, this series diverges.

For 8 + 4 + 2 + 1 + …, divide 4 by 8 to get 1/2. Check: 2 ÷ 4 = 1/2. So r = 1/2. Since |1/2| < 1, this series converges.

One caution: confirm the ratio holds between more than one pair of terms. If it changes, the series is not geometric, and this test does not apply. A single ratio computed from two terms is not enough to classify a series.

What Happens When the Ratio Is Negative?

A negative ratio does not change the test. Use the absolute value.

Consider 1 – 1/2 + 1/4 – 1/8 + … Here r = -1/2. The terms alternate in sign, flipping positive, negative, positive. But |r| = 1/2, which is less than 1, so the series converges. The alternating signs do not prevent the total from settling; they just make it approach its limit from both sides.

Now consider 1 – 2 + 4 – 8 + … Here r = -2, and |r| = 2, which is at least 1. The terms grow and flip sign, so the series diverges. The running total swings wildly and never settles.

This is why the test uses absolute value. The sign of r affects how the series behaves along the way, but it does not change whether the sum converges. Only the magnitude of r decides that.

What Is the Sum When a Geometric Series Converges?

When |r| < 1, the infinite sum has a fixed value given by the formula S = a / (1 - r), where a is the first term.

For 1 + 1/2 + 1/4 + 1/8 + …, the first term a = 1 and r = 1/2. So S = 1 / (1 – 1/2) = 1 / (1/2) = 2. The infinite series adds up to exactly 2.

For 8 + 4 + 2 + 1 + …, a = 8 and r = 1/2. So S = 8 / (1 – 1/2) = 8 / (1/2) = 16.

This formula only works when the series converges. If |r| ≥ 1, there is no finite sum, and plugging into the formula gives a meaningless result. The formula describes where the sum lands, not where it would land if it never landed at all.

Why the Boundary Case r = 1 and r = -1 Diverges

When |r| equals exactly 1, the series diverges, and it is worth seeing why since this is the edge of the test.

If r = 1, every term is identical. The series 5 + 5 + 5 + 5 + … keeps adding 5 forever, so the total grows without bound. It diverges.

If r = -1, the terms alternate between a value and its opposite. The series 5 – 5 + 5 – 5 + … never settles. The partial sums bounce between 5 and 0 forever and approach no single value, so it diverges.

Both cases fall under the |r| ≥ 1 rule. There is no special exception at the boundary; the same test handles them.

A Quick Comparison of Cases

Notice that the sign of r never decides the outcome on its own. Two series with r = 1/2 and r = -1/2 both converge. Two series with r = 2 and r = -2 both diverge. The magnitude is what counts.

Common Mistakes to Avoid

Most errors with this test come from a few predictable places.

• Forgetting the absolute value. A ratio of -3 is not less than 1 in the way that matters. Use |r| = 3, which is at least 1, so the series diverges.
• Assuming a series is geometric when it is not. If the ratio between consecutive terms changes, the test does not apply. Confirm the ratio is constant first.
• Using the sum formula on a divergent series. S = a / (1 – r) only gives a real answer when |r| < 1. Applying it elsewhere produces a number that means nothing.
• Treating r = 1 or r = -1 as a middle case. These are not borderline. They diverge, just like any ratio with absolute value of 1 or more.

One clarification that helps: the test tells you whether the sum settles, not how fast. A series with r = 0.99 converges, but slowly. A series with r = 0.01 converges almost immediately. Both pass the test, but the partial sums behave very differently along the way.

How This Test Fits Into Broader Series Work

The geometric series test is one of the few that gives a clean yes-or-no answer without further work. Many convergence tests for other kinds of series, such as the ratio test used in calculus, rely on the same underlying idea and often reduce to a geometric comparison.

That is why this test is taught early. It anchors the intuition that a series converges when its terms shrink fast enough, and the geometric case is the clearest example of that principle. When terms shrink by a fixed fraction each step, the total is guaranteed to stay bounded. When they do not, it is not.