The divergence test, also called the nth-term test, is one of the first checks to apply to an infinite series. It asks whether the individual terms a_n get closer and closer to 0 as n grows without bound. This matters because a series cannot settle to a finite sum if its added terms fail to shrink to zero.
The test is quick, powerful, and often eliminates obviously divergent series before using more advanced tests.
For an infinite series sum a_n, if lim as n approaches infinity of a_n is not 0, or if that limit does not exist, then the series diverges. If lim as n approaches infinity of a_n = 0, the test gives no conclusion, because some such series converge and others diverge. For example, sum 1/n has terms that approach 0 but still diverges, while sum 1/n^2 also has terms that approach 0 and converges.
The divergence test is therefore a one-way test: it can prove divergence, but it can never prove convergence.
Understanding Calculus: The Divergence Test
The key idea comes from partial sums. A partial sum is the total after a fixed number of terms. Each new partial sum differs from the previous one by exactly one new term.
If the partial sums are supposed to settle near one finite number, the gap between consecutive partial sums must become tiny. That gap is the term being added. This explains why terms that stay near a nonzero value make convergence impossible.
Adding about three every time, for instance, keeps pushing the running total upward. Adding about negative two every time keeps pushing it downward.
Terms can fail the test in less obvious ways. Consider a series whose terms alternate between one and negative one. The terms do not approach any single number.
Its partial sums jump back and forth between one and zero, so they never settle. A term pattern can also contain occasional large spikes. For example, a sequence might be zero for many positions but equal to one whenever the index is a square number.
Even though many terms are zero, the sequence does not approach zero because the ones keep returning forever. The series must diverge. A limit describes what happens eventually, not what happens in only the first few terms.
When using the test, students should examine the term itself before thinking about the whole sum. Simplify fractions, separate factors, and use familiar limit rules. For a rational expression in the index, compare the highest powers in the numerator and denominator.
If the degrees are equal, the terms often approach the ratio of the leading coefficients, which is nonzero. If the numerator has the higher degree, the terms may grow without bound. Either result proves divergence immediately.
Exponential factors need careful attention. A positive base greater than one usually makes terms grow, while a base between zero and one makes them shrink.
Alternating signs do not automatically create convergence. The size of the terms still matters.
The test is best treated as a fast screening step. It saves time before ratio, root, comparison, integral, or alternating series tests are used. In a homework problem, write the limit of the general term first.
If it is nonzero, state that the series diverges by the divergence test. If the limit does not exist, state the same conclusion and briefly identify the oscillation or repeated spikes. If the limit is zero, stop claiming anything from this test.
That result only says convergence remains possible. The next step depends on the form of the series. Recognizing this boundary prevents a common error, which is confusing a necessary condition for convergence with a sufficient condition.
Key Facts
- Divergence test: If lim n -> infinity a_n != 0, then sum a_n diverges.
- If lim n -> infinity a_n does not exist, then sum a_n diverges.
- If sum a_n converges, then lim n -> infinity a_n = 0.
- The converse is false: lim n -> infinity a_n = 0 does not guarantee that sum a_n converges.
- Harmonic series example: sum 1/n diverges even though lim n -> infinity 1/n = 0.
- Geometric example: sum r^n converges for |r| < 1, and its terms satisfy lim n -> infinity r^n = 0.
Vocabulary
- Infinite series
- An infinite series is a sum of infinitely many terms, written in the form sum a_n.
- nth term
- The nth term a_n is the general term of a sequence or series, expressed as a formula involving n.
- Divergence
- Divergence means that an infinite series does not approach a finite total sum.
- Limit of a sequence
- The limit of a sequence is the value that its terms approach as n becomes infinitely large.
- Inconclusive test
- An inconclusive test result means the test does not decide whether the series converges or diverges.
Common Mistakes to Avoid
- Claiming convergence when lim n -> infinity a_n = 0 is wrong because the divergence test cannot prove convergence. A series such as sum 1/n has terms approaching 0 but still diverges.
- Forgetting to check whether the term limit exists is wrong because a nonexisting limit also proves divergence. If a_n oscillates without approaching one value, the series cannot converge.
- Applying the test to partial sums instead of terms is wrong because the divergence test uses a_n, not S_n. The question is whether the added terms approach 0, not whether a running total looks stable at first.
- Using the divergence test after algebra mistakes in a_n is wrong because the conclusion depends completely on the correct term formula. Simplify rational, exponential, or trigonometric expressions carefully before taking the limit.
Practice Questions
- 1 Use the divergence test on the series sum from n = 1 to infinity of (3n + 2)/(5n - 1). Does the series diverge, or is the test inconclusive?
- 2 Use the divergence test on the series sum from n = 1 to infinity of n/(n^2 + 4). What is lim n -> infinity a_n, and what conclusion can you make?
- 3 A student says that because lim n -> infinity 1/n = 0, the series sum 1/n must converge. Explain the error and state what the divergence test actually tells you.