Ratio Test for Convergence

What the Ratio Test Is For

When you're given an infinite series, one of the first questions is whether the series converges or diverges. Some series are easy to classify using a geometric-series test, a p-series, or the comparison tests. Others contain factorials, powers, or complicated products that make those methods awkward.

The Ratio Test gives you another option. It compares the size of consecutive terms by looking at the ratio

If the ratio approaches a number less than 1, the terms are shrinking quickly enough for the series to converge. If the ratio approaches a number greater than 1, the terms are not shrinking fast enough, and the series diverges.

The Ratio Test Formula

For a series

compute the limit

Then compare the result with 1:

If , the series converges absolutely.
If , the series diverges.
If , the Ratio Test is inconclusive.

The absolute value is important. A series can alternate signs, so we care about the magnitude of the ratio when applying the test.

A Step-by-Step Method

Identify the general term of the series.
Write the next term .
Form the ratio .
Simplify the ratio as much as possible before taking the limit.
Take the limit as approaches infinity.
Compare the result with 1 and state clearly whether the series converges, diverges, or cannot be determined by the Ratio Test.

Worked Examples

Example 1: Factorials

Determine whether the series converges or diverges.

Step 1: Identify the general term:

Step 2: Form the ratio:

Step 3: Simplify using :

Step 4: Take the limit:

Conclusion: Since the limit is less than 1, the series converges absolutely.

Example 2: Exponential Terms

Determine whether the series converges or diverges.

Step 1: Let .

Step 2: Form the ratio:

Step 3: Simplify:

Step 4: Take the limit:

Conclusion: Because , the series converges absolutely.

Example 3: Alternating Signs

Determine whether the series converges or diverges using the Ratio Test.

Step 1: Let .

Step 2: Form the ratio and take the absolute value:

Step 3: Simplify:

Step 4: Take the limit:

Conclusion: Since the limit is less than 1, the series converges absolutely. The alternating signs do not change the conclusion.

Example 4: When the Ratio Test Is Inconclusive

Consider the series .

Step 1: Let .

Step 2: Form the ratio:

Step 3: Take the limit:

Conclusion: Because the limit equals 1, the Ratio Test gives no conclusion. This does not mean the series diverges. It means you need another test.

Why Factorials Make the Ratio Test So Useful

The Ratio Test is particularly powerful when factorials appear in the terms of a series. Factorials contain products of many consecutive integers, so writing the ratio often creates massive cancellation.

For example, if a term contains , then the next term contains . Their ratio simplifies dramatically because

This is one of the biggest clues that the Ratio Test may be the best choice: factorials, exponential powers, and products involving often simplify beautifully when you divide consecutive terms.

Reading the Result Correctly

The most important thing to remember is that the Ratio Test has three outcomes, not two.

Students sometimes see and incorrectly conclude that the series diverges. That's not correct. The test simply has no information when the limit equals 1.

Mistake 1: Forgetting the Absolute Value

The Ratio Test uses . If the series alternates signs, the ratio itself may be negative, but convergence is determined by the magnitude of the ratio.

Mistake 2: Treating L = 1 as Divergence

When , the Ratio Test is inconclusive. You must use another convergence test. Some series with converge, while others diverge.

Mistake 3: Not Simplifying Factorials

Don't try to evaluate huge factorials numerically. Instead, cancel them algebraically. For example, simplifies to .

Mistake 4: Giving an Incomplete Conclusion

Don't stop after finding the limit. Always explicitly compare the limit with 1 and state the conclusion: converges absolutely, diverges, or the test is inconclusive.

Ratio Test vs. Other Convergence Tests

The Ratio Test is not always the fastest method. Choosing the right test is part of mastering infinite series.

Geometric series: If the series has the form , use the geometric-series test directly.
p-series: If the series looks like , use the p-series rule.
Comparison Test: Useful when the series resembles a familiar benchmark series.
Limit Comparison Test: Useful when a complicated positive-term series behaves like a simpler known series.
Alternating Series Test: Useful when terms alternate signs and their magnitudes decrease toward zero.
Ratio Test: Especially useful for factorials, exponentials, powers, and products.

Practice Problems

Question 1

Use the Ratio Test to determine whether converges or diverges.

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Question 2

Use the Ratio Test to determine whether converges or diverges.

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Question 3

Use the Ratio Test to determine whether converges or diverges.

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Question 4

Apply the Ratio Test to . If the test is inconclusive, state that explicitly.

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AP Calculus Exam Strategy

On an AP Calculus BC problem, don't mechanically use the Ratio Test just because you know it. Look at the structure of the series first.

If you see factorials, exponential powers, or expressions such as , the Ratio Test should immediately come to mind. The consecutive-term ratio often produces cancellation that makes the limit almost automatic.

And remember the critical decision point:

If you get , stop. The Ratio Test has done all it can do. Move to another test rather than trying to force a conclusion.

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