The divergence test is a quick way to check whether an infinite series diverges. Also called the nth term test, it says: if the terms of a series don’t shrink toward zero, the series cannot possibly converge. It’s typically the first test you apply when analyzing any infinite series, because it takes seconds and can save you from wasting time on more complex methods.
How the Test Works
The logic is straightforward. Given a series with terms a_n, compute the limit of a_n as n approaches infinity. If that limit is anything other than zero, the series diverges. This includes cases where the limit equals some nonzero number and cases where the limit doesn’t exist at all (for instance, if the terms oscillate without settling down).
The intuition behind it: when you’re adding up infinitely many numbers, the only way the running total can stabilize is if the numbers you’re adding get vanishingly small. If each new term keeps contributing a meaningful amount, the sum will grow without bound or bounce around forever. It will never settle on a finite value.
Applying the Test Step by Step
To use the divergence test on a series, you only need to do two things:
- Find the limit. Take the general term a_n and evaluate its limit as n goes to infinity. Use whatever limit techniques you already know: dividing by the highest power, L’Hôpital’s rule, squeeze theorem, etc.
- Interpret the result. If the limit is not zero (or doesn’t exist), the series diverges. If the limit equals zero, the test tells you nothing, and you need a different test.
That’s it. There’s no further calculation involved, which is why this test is always worth trying first.
Examples Where the Test Proves Divergence
Consider the series whose general term is n³ + 2 divided by n² + 1. The degree of the numerator (3) is higher than the degree of the denominator (2), so the limit as n approaches infinity is infinity. Since that’s not zero, the divergence test immediately confirms the series diverges. No other test needed.
A slightly trickier example: the series with terms 2k/(k + 5). As k grows, this expression approaches 2, not zero. The series diverges.
Here’s one that catches people off guard: the series with terms (-1)^n cos(1/n). As n goes to infinity, 1/n shrinks to zero, and cos(0) = 1. So the terms alternate between values approaching +1 and -1. They never approach zero. The divergence test confirms this series diverges, even though the alternating sign might make it look like the terms are canceling out.
The Biggest Mistake Students Make
The divergence test is a one-way tool. It can prove divergence, but it can never prove convergence. This is the single most common error in working with this test: finding that the limit of a_n equals zero and concluding that the series converges. That conclusion is not valid.
The harmonic series is the classic counterexample. Its terms are 1/n, and the limit of 1/n as n approaches infinity is zero. The divergence test is therefore inconclusive. And yet the harmonic series diverges, a fact you can prove with other tests like the integral test. So even when terms shrink to zero, the series can still diverge if the terms don’t shrink fast enough.
Meanwhile, the series with terms 1/n² also has a limit of zero, and this series does converge. Both series pass the divergence test (meaning the test is inconclusive for both), yet one converges and the other doesn’t. This is exactly why a limit of zero tells you nothing on its own.
Why the Logic Only Runs One Direction
If a series converges, then its terms must approach zero. That’s a proven theorem. The divergence test is simply the contrapositive of that statement: if the terms don’t approach zero, the series can’t converge. Contrapositives are always logically equivalent to the original statement, so the test is rock solid in that direction.
But the converse, “if the terms approach zero, the series converges,” is a different statement entirely, and it’s false. Knowing the terms shrink is a necessary condition for convergence, not a sufficient one. Think of it like this: having fuel is necessary for a car to run, but having fuel alone doesn’t guarantee the car will start.
Where the Divergence Test Fits Among Other Tests
In a typical calculus course, you’ll learn a whole toolkit of convergence tests: the ratio test, the comparison test, the integral test, the alternating series test, and others. The divergence test sits at the top of the decision tree. It’s always the first thing to check, because it’s the simplest and fastest. If the terms clearly don’t go to zero (polynomials where the numerator grows faster than the denominator, exponential terms, oscillating sequences), you’re done in one step.
If the divergence test comes back inconclusive, meaning the limit is zero, that’s your signal to move on to a more powerful test. The ratio test works well for series involving factorials or exponentials. The comparison and limit comparison tests are useful when you can relate your series to a known benchmark like a p-series. The integral test handles terms that look like functions you can integrate. None of these replace the divergence test; they pick up where it leaves off.
Getting in the habit of always running the divergence test first will save you time on exams and homework. There’s no reason to set up an elaborate ratio test calculation if the terms of your series approach 3 instead of zero.