The Divergence Test (nth Term Test)

You have a series, and you need to know if it sums to a finite number. The first test to reach for is the divergence test. It is the fastest check in calculus II and the only one that never requires a second opinion, as long as it gives you a verdict.

The divergence test (also called the nth term test) has a simple instruction: If the limit of the nth term a_n as n goes to infinity is not zero, then the series Σ a_n diverges. That is it. One limit, one answer.

Stewart, Calculus, section 11.2 gives the formal version: If lim_{n→∞} a_n ≠ 0, then Σ a_n diverges. If lim_{n→∞} a_n = 0, the test is inconclusive, the series may converge or diverge. That zero case is where students get into trouble.

A series like Σ n/(n+1) has a_n = n/(n+1). The limit of n/(n+1) as n→∞ is 1. Because 1 ≠ 0, the series diverges. That is a complete answer in under ten seconds.

Now consider Σ 1/n, the harmonic series. The limit of 1/n is 0. The divergence test gives no verdict. The harmonic series actually diverges, but you cannot learn that from this test, you need a p-series test or the integral test to prove it.

The biggest mistake students make with the nth term test is assuming that if the limit is zero, the series converges. That is not true. The harmonic series Σ 1/n has a limit of zero for its terms, yet it diverges. This series is the classic counterexample.

The harmonic series is a p-series with p=1. The p-series test tells you that Σ 1/n^p converges only when p > 1. At p=1, the series diverges. So the limit of terms being zero is a necessary condition for convergence, not a sufficient one.

Run the Cheapest Test First

Run the divergence test before any other test because it is the cheapest option. It requires one limit calculation, a skill from calculus I, and it takes under 30 seconds. If the limit is not zero, you are done. If the limit is zero, you now know which tests to skip and which to try.

The ratio test, the root test, the integral test, the comparison test, all of them are slower. The divergence test is the only one that can give you a definitive 'diverges' answer from a single limit. When you are under exam time pressure, this is the first tool you use.

What to Do When the Test Is Inconclusive

If the limit is zero, the nth term test is inconclusive. That is when you move to the next test in your hierarchy: check if the series is geometric (constant ratio r) or a p-series (1/n^p). If neither fits, try the ratio test for factorials or exponentials, or the comparison test for rational functions.

The divergence test is decisive only when it detects divergence. A limit of zero tells you nothing. Students often claim convergence when they see limit = 0, but that is a logical error. The test says nothing about convergence in that case, it is inconclusive.

For example, Σ 1/n^2 has a limit of zero for its terms. The divergence test is silent. You must use a different test, the p-series test shows convergence because p=2 > 1. The Basel problem gives the exact sum: Σ 1/n^2 = π²/6, but the divergence test cannot give you that sum. No convergence test does; they only tell you if a sum exists.

One common mistake is applying the nth term test to a geometric series and stopping after checking the limit. A geometric series like Σ (1/2)^n has a term limit of 0, so the divergence test is inconclusive. The series actually converges because |r| = 1/2 < 1. Do not skip the geometric series test just because the divergence test gave no answer.

Another mistake is confusing the divergence test with the integral test. The integral test requires a positive, decreasing function and compares the series to an integral. The divergence test only checks the limit of the terms. They are different tools with different conditions.

The most common failure with the divergence test is stopping at limit = 0 and declaring convergence. That is not what the test says. The test is a one-way gate: it only allows you to leave, not to enter. When the limit is zero, you have learned nothing about convergence, and you must keep testing.