Alternating Series Test for Convergence
Check eventually decreasing magnitudes and a zero limit before using alternating-series convergence.
College Board Topic 10.7 · CED p. 187 (PDF p. 192) · CED effective Fall 2020; project course outline 2026–27.
For or , with bₙ nonnegative, the alternating-series test proves convergence when bₙ eventually decreases and . A finite prefix does not matter. Alternation alone is not sufficient, and a zero term limit without the decreasing-size condition does not make this test applicable.
The reason is a squeeze. Even and odd partial sums approach from opposite sides; their gap is the next term magnitude, which tends to zero. This gives convergence without forcing the absolute-value series to converge. Later we will use the same picture for an error bound.
For , the magnitudes decrease to zero. Thus converges. Its absolute-value series is a p-series with p=1/2 and diverges, so the convergence is conditional, a classification developed in Topic 10.9.
When a proposed alternating sequence has irregular magnitudes, do not silently claim monotonicity. Check an inequality or derivative, and distinguish the test being inconclusive from the series being divergent. The retained alternating-series lesson supplies more reviewed practice.
Let's work through it
- For , set . Why: Separate signs from magnitudes.
- The magnitudes decrease and tend to zero. Why: n↦n⁻¹ᐟ² is decreasing and has limit zero.
- The series converges by the alternating-series test. Why: All the hypotheses are met, not just the alternating signs.
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
Its magnitudes stay at 1 instead of tending to zero.
Retrieval 2
Worked answer and reason
A finite prefix cannot change convergence.
Catch the tempting mistake
- Alternating signs guarantee convergence.
- You also need magnitudes tending to zero and a suitable eventually decreasing tail for this test.
Calculator workspace and limits
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Put it to work
- Unit 10 problem 13 · No calculator · analytic
- Unit 10 problem 14 · No calculator · verbal
For the teacher
- Ask learners to identify bₙ explicitly. If the magnitude is not decreasing, ask what another method might show rather than declaring divergence from a failed test.
Glossary and deeper reading · 7 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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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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Partial sum
A partial sum adds the first finitely many terms of a series. The nth partial sum is Sₙ=a₁+a₂+⋯+aₙ when indexing begins at 1. For example, the first three terms of 1/2+1/4+1/8+⋯ give partial sum 7/8. Convergence of a series concerns the limit of these sums, not just the limit of the individual terms.
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Error bound
An error bound gives a justified upper limit for the magnitude of an approximation’s error. It states how far the approximate value can be from the true value. For example, an error bound of 0.01 places the true value within 0.01 of the approximation. A bound requires its stated hypotheses and is not a claim that the error actually equals the bound.
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p-series
A p-series has terms 1/nᵖ for positive integers n. It converges exactly when p>1. For example, the p-series with p=2 converges, while p=1 gives the divergent harmonic series. The boundary p=1 belongs to the divergent cases.
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Sequence
A sequence is an ordered list of terms indexed by integers. A rule can specify each term directly or recursively from earlier terms. For example, aₙ=1/n produces 1, 1/2, 1/3, and so on. An infinite sequence converges when its terms approach a finite limit as the index grows without bound.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.