Ratio Test for Convergence
Use absolute successive-term ratios and switch tests when the limit is one or does not exist.
College Board Topic 10.8 · CED p. 188 (PDF p. 193) · CED effective Fall 2020; project course outline 2026–27.
For a series with nonzero terms on a tail, compute . If , the series converges absolutely. If or the ratio grows without bound, the series diverges. If , the test is inconclusive. If the ratio limit does not exist, this limit-based test gives no conclusion.
The absolute value is part of the test. For L<1, the tail is eventually dominated by a convergent geometric series. For L>1, the term magnitudes eventually grow rather than tend to zero. Factorials and exponentials are especially friendly because their ratios cancel cleanly.
For , the ratio is , so the series converges. For 1/n and 1/n², both successive-term ratios tend to 1, even though the first sum diverges and the second converges. This is why L=1 must trigger a new test.
The retained ratio-test lesson is aligned practice. Its neighboring root-test lesson is enrichment: the current BC CED specifically lists the assessed tests without the root test. Know the assessed methods well enough to choose among them.
Let's work through it
- For , form . Why: Expand (n+1)!=(n+1)n! and cancel matching factors.
- The ratio tends to . Why: The denominator grows without bound.
- The series converges absolutely. Why: The ratio-test convergence condition holds.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Worked answer and reason
It is inconclusive; both convergent and divergent examples have L=1.
Retrieval 2
Worked answer and reason
A negative signed ratio does not itself establish the necessary geometric-size bound.
Catch the tempting mistake
- L=1 means the series diverges.
- Use another test. The p-series 1/n² converges despite ratio limit 1.
Calculator workspace and limits
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Scientific calculator
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Put it to work
- Unit 10 problem 15 · No calculator · analytic
- Unit 10 problem 16 · No calculator · analytic
For the teacher
- Use factorial cancellation exercises, then deliberately include harmonic and p=2 examples so students practice abandoning an inconclusive ratio test.
Glossary and deeper reading · 5 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Infinite series
An infinite series is the sum of the terms of an infinite sequence, defined through its partial sums. The series converges when those partial sums approach a finite value. For example, 1/2+1/4+1/8+⋯ converges to 1. Terms approaching 0 are necessary for convergence, but that condition alone is insufficient.
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Absolute convergence
A series converges absolutely when the series of its absolute term values converges. Absolute convergence also guarantees convergence of the original signed series. For example, an alternating series with magnitudes 1/n² converges absolutely. An absolutely convergent series retains its sum under any rearrangement of its terms.
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Convergence of a series
A series converges when the sequence of its partial sums approaches a finite limit. It diverges when there is no such finite limit. For example, 1/2+1/4+1/8+⋯ converges to 1, while the harmonic series diverges. Convergence of the individual terms to 0 is necessary but does not by itself establish convergence of their sum.
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Geometric series
A geometric series adds terms that have a fixed common ratio. With first term a and ratio r, it converges to a/(1−r) when |r|<1. For example, 1+1/2+1/4+⋯ sums to 2. For a nonzero first term and |r|≥1, the infinite series diverges.
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Root test
The root test examines the limiting nth root of the absolute value of the nth series term. A limit below 1 gives absolute convergence, and a limit above 1 or infinite gives divergence. For example, terms (1/3)ⁿ have nth-root limit 1/3 and their series converges absolutely. A limit equal to 1 is inconclusive.