The nth Term Test for Divergence

What the nth Term Test Is For

When you are given an infinite series, one of the first questions is whether the series converges to a finite value or diverges. The nth term test, also called the divergence test, gives you one of the quickest ways to prove divergence.

Suppose your series is written as

The nth term test asks a very simple question: what happens to the individual terms as n gets larger and larger?

If the terms do not approach zero, the series cannot possibly converge.

The Key Idea

For an infinite series to converge, its individual terms must approach zero. In other words, if

then convergence is possible.

But if the terms approach a nonzero number, approach infinity, or fail to have a limit at all, the series must diverge.

This is the entire test. The calculation is often just a single limit.

The nth Term Test

For the series

calculate the limit of the individual term:

Then:

• If the limit is not zero, the series diverges.
• If the limit is infinite, the series diverges.
• If the limit does not exist, the series diverges.
• If the limit is zero, the test is inconclusive.

Why Does It Work?

Imagine adding infinitely many terms together. If the terms eventually become tiny, there is at least a possibility that their total can settle toward a finite number.

But suppose the terms look like

They are not getting closer to zero. Every time you add another term, you are still adding a substantial amount. There is no way the partial sums can settle at a finite value.

Even if the terms alternate between positive and negative values, they still need to approach zero. For example, terms such as

do not approach zero. The signs change, but the magnitude remains 1. Therefore, the associated series cannot converge.

A Step-by-Step Method

1. Identify the nth term. Call it .
2. Take the limit as n approaches infinity. Calculate .
3. Check the result. Is the limit zero, nonzero, infinite, or nonexistent?
4. State the conclusion precisely. If the limit is not zero or does not exist, the series diverges by the nth term test.
5. If the limit is zero, stop. The nth term test is inconclusive. You need another convergence test.

Worked Examples

The Critical Case: When the Limit Is Zero

This is where students most often misuse the nth term test.

Consider the harmonic series:

The nth term is

and

So the nth term test tells us nothing about convergence here.

In fact, the harmonic series diverges, but you need another test, such as the integral test, to prove it.

Necessary vs. Sufficient

This distinction is extremely important on AP Calculus BC.

If a series converges, then its terms must approach zero:

Therefore, if the terms do not approach zero, convergence is impossible.

But the reverse statement is not true. Knowing that

does not guarantee that the series converges.

Think of the nth term test as a one-way test:

But you cannot reverse that implication.

When the nth Term Test Is the Right Test

The nth term test is especially useful when the nth term contains something that clearly prevents it from approaching zero.

Look for expressions such as:

• A constant nonzero limit.
• A term that grows without bound.
• An alternating term whose magnitude does not shrink to zero.
• A rational expression whose numerator and denominator have the same degree and therefore approach a nonzero ratio.
• Any expression where the limit of the nth term is clearly not zero.

When you see one of these, the nth term test can often settle the question immediately.

When You Need Another Test

If the nth term approaches zero, you should immediately ask what other test applies.

Common AP Calculus BC convergence tests include:

• Geometric series: Use the geometric-series rule when the series has a constant ratio.
• p-series: Recognize series of the form .
• Integral test: Compare a series with an improper integral.
• Comparison test: Compare the terms with a known convergent or divergent positive series.
• Limit comparison test: Compare the limiting ratio of two positive-term series.
• Alternating series test: Use when signs alternate and the magnitudes decrease toward zero.
• Ratio test: Especially useful for factorials, exponentials, and power-series-type expressions.
• Root test: Particularly useful when the nth term contains an nth root or nth power.

A Useful AP Exam Shortcut

Before trying a complicated convergence test, always check the nth term.

For example, suppose you encounter

You might be tempted to start with the ratio test. But first look at the nth term:

The denominator grows much more slowly than the numerator, so

The series diverges immediately.

There is no reason to spend time applying a more complicated test once the nth term test has already settled the question.

Common Mistakes

Practice Problems

The AP Calculus BC Takeaway

The nth term test is one of the simplest convergence tests, but it is also one of the easiest to misuse.

Remember this three-part rule:

The first two cases prove divergence. The third case does not prove convergence—it simply means you need another test.

On the AP exam, checking the nth term should often be one of your first moves when you are asked whether an infinite series converges or diverges. If the terms fail to approach zero, you can finish the problem immediately.