Ratio Test for Convergence
Revisit the ratio test to find an open power-series convergence interval before testing endpoints.
Before this connection: Series convergence and alternating hypotheses.
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
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
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.