The nth Term Test for Divergence
Use a nonzero or missing term limit to prove divergence and recognize that a zero limit is inconclusive.
College Board Topic 10.3 · CED p. 183 (PDF p. 188) · CED effective Fall 2020; project course outline 2026–27.
If converges, then . You can see why by writing : if both partial sums approach the same finite limit, their difference approaches zero. Therefore a nonzero term limit, or a term limit that does not exist, proves divergence.
The converse is false. A zero term limit tells you only that this particular divergence test has not ruled the series out. For example, 1/n tends to zero, yet the harmonic series diverges. Geometric terms with ratio 1/2 also tend to zero, and their series converges. You need another test after a zero term limit.
Use this quick test before doing long algebra. For , the term limit is 1, so the series diverges immediately. A numeric display that shows terms close to 1 is suggestive; dividing numerator and denominator by n establishes the actual limit.
Let's work through it
- For , examine . Why: The nth-term test inspects the summand, not the running total.
- Divide by n to obtain . Why: This is an exact limit calculation.
- The series diverges. Why: A convergent series must have terms tending to zero.
Try it from memory
Pause & retrieve
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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
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Retrieval 1
Worked answer and reason
The necessary zero-term condition fails.
Retrieval 2
Worked answer and reason
The condition is necessary, not sufficient.
Catch the tempting mistake
- The nth-term test proves convergence when the term limit is zero.
- It is a divergence test. The harmonic series is a counterexample to that claimed converse.
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Put it to work
- Unit 10 problem 5 · No calculator · analytic
- Unit 10 problem 6 · No calculator · verbal
For the teacher
- Require the words inconclusive by this test when the limit is zero. Do not accept a checkmark labeled converges.
Glossary and deeper reading · 4 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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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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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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Harmonic series
The harmonic series is 1+1/2+1/3+1/4+⋯. It diverges even though its individual terms approach 0. For example, grouping terms from 1/2 through 1/3+1/4 and then 1/5+⋯+1/8 shows repeatedly added blocks of at least 1/2. It is a standard comparison that separates a necessary term condition from actual convergence.