The nth Term Test for Divergence

students, imagine you are watching a never-ending list of numbers and asking a simple question: does the total settle down to a finite value, or does it keep growing without limit? 📈 In AP Calculus BC, one of the first tools for answering that question is the nth Term Test for Divergence. This test is quick, useful, and often the first checkpoint when you study an infinite series.

Learning Objectives

By the end of this lesson, students, you should be able to:

explain the meaning of the nth Term Test for Divergence,
use it to decide when a series must diverge,
connect it to the bigger picture of infinite sequences and series,
recognize what the test can and cannot prove,
use examples to justify conclusions clearly on AP Calculus BC work.

The main idea is simple: if the terms of a series do not approach $0$, then the series cannot converge. That sounds small, but it is powerful. However, if the terms do approach $0$, the series may still diverge, so this test is only a starting point, not a final answer.

What a Series Is and Why the Last Terms Matter

A sequence is an ordered list of numbers like $a_1, a_2, a_3, \dots$. A series is the sum of those terms:

$$\sum_{n=1}^{\infty} a_n$$

A series tries to add infinitely many terms. That may sound impossible, but in calculus we ask whether the partial sums approach a finite limit. The partial sum is

$$S_N = \sum_{n=1}^{N} a_n$$

If the sequence of partial sums $\{S_N\}$ approaches a finite number, then the series converges. If not, it diverges.

The nth Term Test looks at the terms $a_n$ themselves. Why? Because if the total sum is going to settle down, then the pieces being added must become tiny. In fact, for a series to converge, it is necessary that

$$\lim_{n\to\infty} a_n = 0$$

If this limit is not $0$, or does not exist, then the series diverges. This is the core of the test.

The nth Term Test for Divergence

The test can be stated like this:

$$\lim_{n\to\infty} a_n \neq 0$$

or if the limit does not exist, then

$$\sum_{n=1}^{\infty} a_n \text{ diverges}$$

This is often called the divergence test or the nth term test.

Notice the careful wording. The test tells you when a series definitely diverges. It does not tell you that a series converges just because the terms go to $0$.

That difference is extremely important. For example, the harmonic series

$$\sum_{n=1}^{\infty} \frac{1}{n}$$

has terms that satisfy

$$\lim_{n\to\infty} \frac{1}{n} = 0$$

but the series still diverges. So passing the nth Term Test only means you need another test to keep investigating. 🧠

A good way to remember the test is:

if the terms do not go to $0$, the series must diverge,
if the terms do go to $0$, the test gives no final answer.

Examples That Show the Test in Action

Let’s look at several examples, because AP Calculus BC often asks you to justify results using a limit.

Example 1: A geometric-like term that does not go to $0$

Consider

$$\sum_{n=1}^{\infty} \frac{n}{n+1}$$

Here the terms are

$$a_n = \frac{n}{n+1}$$

Now compute the limit:

$$\lim_{n\to\infty} \frac{n}{n+1} = 1$$

Because the limit is not $0$, the series diverges by the nth Term Test.

This is a strong example because the terms are close to $1$ for large $n$, so adding them forever would keep adding numbers near $1$. That cannot settle to a finite sum.

Example 2: Terms that oscillate

Consider

$$\sum_{n=1}^{\infty} (-1)^n$$

The terms are

$$a_n = (-1)^n$$

These terms switch between $-1$ and $1$, so the limit

$$\lim_{n\to\infty} (-1)^n$$

does not exist. Since the limit is not $0$, the series diverges by the nth Term Test.

This example shows that a series can diverge even if its terms do not grow larger. Oscillation alone can prevent convergence.

Example 3: Terms that do go to $0$, but the series still diverges

Consider the harmonic series again:

$$\sum_{n=1}^{\infty} \frac{1}{n}$$

The limit of the terms is

$$\lim_{n\to\infty} \frac{1}{n} = 0$$

So the nth Term Test does not prove divergence. But it also does not prove convergence. In fact, the series diverges, but you need another test, such as the $p$-series test with $p=1$.

This example is a classic AP trap. Students sometimes think “the terms go to $0$” means “the series converges.” That is false. The nth Term Test only gives a one-way conclusion.

How to Use the Test on AP Calculus BC

On the AP exam, you may be asked to determine whether a series converges or diverges, or to justify your answer in a sentence or two. The nth Term Test is one of the fastest tools available.

A good procedure is:

Identify the general term $a_n$.
Compute

$$\lim_{n\to\infty} a_n$$

If the limit is not $0$ or does not exist, write that the series diverges by the nth Term Test.
If the limit is $0$, say the test is inconclusive and use another method.

For example, suppose you are given

$$\sum_{n=1}^{\infty} \frac{2n+5}{3n-1}$$

Then

$$\lim_{n\to\infty} \frac{2n+5}{3n-1} = \frac{2}{3}$$

Since this is not $0$, the series diverges by the nth Term Test.

Another example:

$$\sum_{n=1}^{\infty} \frac{1}{n^2}$$

Here

$$\lim_{n\to\infty} \frac{1}{n^2} = 0$$

So the test gives no conclusion. You would need another method, such as recognizing it as a $p$-series with $p=2$.

What the Test Does Not Do

A major strength of the nth Term Test is its simplicity, but its weakness is just as important. It can only prove divergence in a specific way.

It cannot:

prove convergence,
find the exact sum of a series,
tell whether a series with terms approaching $0$ converges or diverges.

That is why the nth Term Test is often the first step, not the last one. If it fails to show divergence, you move on to other tools such as:

the geometric series test,
the $p$-series test,
the integral test,
the comparison tests,
the alternating series test,
the ratio or root test.

In the larger unit on Infinite Sequences and Series, this test helps you quickly eliminate impossible cases. It acts like a screening tool. If the terms do not vanish, the series is already finished: it diverges.

Real-World Connection

Think about saving money in a piggy bank 🐷. If every deposit is $1$, then the total clearly grows forever. That is like a series whose terms do not approach $0$. Even if the deposits are alternating between $1$ and $-1$, the total never settles down.

Now imagine your deposits get smaller and smaller, like $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, and so on. The terms do approach $0$, so the nth Term Test does not immediately reject the series. But whether the total stays finite depends on the pattern of the deposits. That is why other tests matter.

This is a useful mindset for AP Calculus BC: first check the size of the terms, then decide whether more analysis is needed.

Conclusion

The nth Term Test for Divergence is one of the most important starting points in the study of infinite series. It says that if

$$\lim_{n\to\infty} a_n \neq 0$$

or if the limit does not exist, then

$$\sum_{n=1}^{\infty} a_n$$

diverges. This test is fast, reliable, and easy to apply, but it only proves divergence. If the terms do approach $0$, the series may still converge or diverge, so you must continue with another test.

For AP Calculus BC, students, remember the key idea: a convergent series must have terms that go to $0$, but terms going to $0$ do not guarantee convergence. That distinction is essential and appears often on exams. ✅

Study Notes

The nth Term Test for Divergence checks the limit of the terms $a_n$ in a series.
If

$$\lim_{n\to\infty} a_n \neq 0$$

or the limit does not exist, then

$$\sum_{n=1}^{\infty} a_n$$

diverges.

$$\lim_{n\to\infty} a_n = 0$$

the test is inconclusive.

The test can prove divergence, but it cannot prove convergence.
The harmonic series

$$\sum_{n=1}^{\infty} \frac{1}{n}$$

is a classic example where the terms go to $0$ but the series still diverges.

Always identify the general term $a_n$ before applying the test.
If the nth Term Test does not settle the question, use another convergence test.