The nth Term Test for Divergence
Introduced| English |
|---|
| nth Term Test/ˌenˌtiːˈeɪtʃ tɜːm test/ |
A quick way to prove divergence
- Before running a fancy test, do the cheapest check: look at the terms.
- The nth Term Test 通项判别法 (or Divergence Test) uses the limit of the terms $a_n$.
- If the terms don't shrink to $0$, the series can't converge — it diverges immediately.
- It's the first thing to try, and it only ever proves divergence.
The test
- Compute $\displaystyle\lim_{n\to\infty} a_n$.
- If $\displaystyle\lim_{n\to\infty} a_n \neq 0$ (or the limit doesn't exist), the series $\sum a_n$ diverges.
- Intuition: if you keep adding pieces that don't shrink to $0$, the total can never settle.
- One limit computation, and you may be done.
Terms that don't reach zero
If a series' terms level off at a nonzero value, the sum can never settle — the nth Term Test proves divergence.
If $\lim_{n\to\infty}a_n\neq 0$, then $\sum a_n$...
Nonzero term limit → divergence.
It can never prove convergence
- If $\displaystyle\lim_{n\to\infty} a_n = 0$, the test is inconclusive — it tells you nothing.
- The series might converge or diverge; you must use another test.
- The harmonic series $\sum\tfrac1n$ has $a_n\to0$ but diverges — proof the test can't confirm convergence.
- So a "$0$" limit is a dead end for this test, not a green light.
If $\lim_{n\to\infty}a_n=0$, the nth Term Test is...
It only proves divergence; a $0$ limit tells you nothing.
The nth Term Test can prove a series converges.
It can only prove divergence.
The series $\sum\tfrac1n$ has $a_n\to0$ but diverges. This shows the nth Term Test...
A $0$ limit doesn't guarantee convergence.
Where it fits in the strategy
- Always run the nth Term Test first — it's fast and can save you a hard test.
- Terms not $\to0$? Done: diverges.
- Terms $\to0$? Move on to a real convergence test (geometric, integral, comparison, ratio, alternating).
- It's a filter, not a full answer.
For $\sum\dfrac{n}{2n+1}$, $\lim a_n=\tfrac12$. So the series...
$\tfrac12\neq0$ → diverges.
As a fast filter, run the nth Term Test ____ among the convergence tests.
It's the cheapest divergence check.
The nth Term Test can only prove divergence. If $\lim a_n=0$, it is inconclusive — do not conclude convergence. The single most common misuse is "$a_n\to0$, so it converges" — wrong; $\sum\frac1n$ is the counterexample.
Does $\displaystyle\sum_{n=1}^{\infty}\dfrac{n}{2n+1}$ converge?
- Terms: $a_n=\dfrac{n}{2n+1}$. Limit: $\displaystyle\lim_{n\to\infty}\dfrac{n}{2n+1}=\dfrac12$.
- Since the limit is $\tfrac12\neq0$, the series diverges by the nth Term Test.
The nth Term Test: if $\lim_{n\to\infty}a_n\neq 0$ (or DNE), then $\sum a_n$ diverges. If $\lim a_n=0$, the test is inconclusive (it never proves convergence). Run it first as a fast divergence filter.