The nth term test cannot prove convergence: terms going to zero is necessary and nowhere near sufficient

01The mistake

Students write “$\lim_{n\to\infty} \frac{1}{n} = 0$, therefore $\sum \frac{1}{n}$ converges.” The harmonic series diverges. The terms do go to zero, and they do so too slowly for the sum to be finite.

The tell is a one-line justification. Any argument for convergence that consists only of a limit of the terms is this error, whatever the series. Convergence requires an actual convergence test — comparison, integral, ratio, alternating series — and the nth term test is not one of them.

It is reinforced by the fact that the conclusion is often right. Most series students meet with terms going to zero do converge, so the wrong reasoning produces correct verdicts frequently. The error is in the justification, and on the exam the justification is what is scored.

It also generates the p-series confusion (U10-CA6). $\sum \frac{1}{n}$ diverges and $\sum\frac{1}{n^2}$ converges, and both have terms tending to zero. A student using only the nth term test cannot tell them apart and will treat the $p > 1$ threshold as an arbitrary rule rather than as the thing that actually decides.

02Why it makes sense to the student

The name invites the misuse. Calling it “the nth term test” sounds like a test that returns a verdict about the series. Its full name — the test for divergence — states the restriction, and the short name that gets used in class quietly removes it.

It is the first test taught and the easiest to apply. Take a limit, compare to zero, done. Every other test requires more work, so under time pressure this is the one that gets reached for whether or not it applies.

Terms shrinking to nothing genuinely feels like it should be enough. If you are adding smaller and smaller amounts, the intuition that the total settles down is strong and almost right — it is a question of how fast, and nothing in the students' experience calibrates that.

And one-directional implications are the recurring failure of the whole course. Continuity and differentiability in Unit 2, critical points and extrema in Unit 5, and this. Students store an association between two conditions and lose which way the arrow points, and it is worth naming that pattern explicitly by the time they reach series.

03The correction

State the test in the only form that is true, and never in the other: if $\lim_{n\to\infty}a_n \neq 0$, the series diverges. That is all it says. If the limit is 0, the test is inconclusive and you must use something else. Write “inconclusive” on the board as the second outcome, because a test with two outcomes rather than three is exactly the misreading.

Then give the harmonic series and prove it diverges, because assertion will not survive the intuition. The grouping argument is short enough for one board: $\frac{1}{3}+\frac{1}{4} > \frac{1}{2}$, then $\frac{1}{5}+\cdots+\frac{1}{8} > \frac{1}{2}$, then the next eight terms exceed $\frac{1}{2}$, and so on without end. Infinitely many half-units, so the sum grows without bound, while every term goes to zero.

Put $\sum\frac{1}{n}$ and $\sum\frac{1}{n^2}$ side by side. Same limiting behaviour of the terms, opposite verdicts. That pairing shows more clearly than any statement that the limit of the terms cannot be what decides, since it is identical in both cases.

Insist on the two-sentence justification: which test, and what it concluded. “The terms go to zero” is never a complete answer for convergence. This is a writing requirement with a concept inside it, and it makes the error impossible to commit silently.

A useful classroom test: “$\lim_{n\to\infty}a_n = 0$. What can you conclude about $\sum a_n$?” The answer is nothing. Students who say it converges have the biconditional; students who say the test is inconclusive have the concept. One question, no computation, and it is worth asking before the other tests are taught rather than after.

04A sample question

05What each wrong answer reveals

• A The implication reversed. The dominant wrong answer, and the reasoning is stated plainly: terms approach zero, therefore convergence. The student has a one-way theorem stored as a two-way one. The harmonic series is the correction and it has to be proved, not asserted — the grouping argument is what displaces the intuition, because the intuition is otherwise very strong.
• B Correct. The nth term test is inconclusive when the limit is 0. The harmonic series is the standard example of a series whose terms tend to zero and which nonetheless diverges, as the grouping argument shows.
• C The test applied without its condition. This student reaches the right verdict for this particular series by an argument that would give the same verdict for every series, including convergent ones. Worth catching precisely because the answer is right — ask what their reasoning would say about $\sum\frac{1}{n^2}$, and the method's failure is immediate.
• D Sum confused with the limit of the terms. This is U10-CA1: the sequence, the partial sums, and the series merged into one object. The student has reported the limit of $a_n$ as the value of $\sum a_n$. It is a more basic confusion than A and needs the three objects separated before any convergence test will be meaningful.

A and C both concern the direction of the implication and fail in opposite ways — A concludes convergence from a limit of zero, C concludes divergence from any limit at all. C is more dangerous in practice because it produces the correct answer here and will produce a wrong one on the next problem. D is a different and earlier confusion. Only B treats the test as inconclusive, which is the outcome the misconception deletes.

06Try it in Mistake Master