Divergence Test

students, in Calculus 2 you will often meet infinite series that look complicated and intimidating at first glance 😅. The good news is that one of the first tools you can use is the Divergence Test, also called the nth-term test for divergence. It is simple, fast, and useful for ruling out series that definitely do not converge.

Lesson goals

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

explain what the Divergence Test says and what its terms mean,
use the test to determine when a series definitely diverges,
understand what the test can and cannot prove,
connect this test to other ideas in Convergence Tests I, such as the integral test and p-series,
recognize why checking the terms of a series is the first step in many problems.

What is a series?

A series is the sum of the terms in a sequence. If a sequence is $\{a_n\}$, then the associated infinite series is

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

This means we add up infinitely many terms like $a_1+a_2+a_3+\cdots$.

A series can either converge or diverge. If the sum approaches a finite number, the series converges. If it does not approach a finite number, it diverges. This is the big question in Convergence Tests I 📘.

The Divergence Test idea

The Divergence Test is based on a very simple observation: if the terms of a series do not get close to $0$, then the series cannot converge.

The test is usually stated like this:

If $\lim_{n\to\infty} a_n \neq 0$, or if the limit does not exist, then

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

diverges.

That is the whole test. It is a necessary condition for convergence that

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

If this does not happen, convergence is impossible.

Why does this make sense?

Think about adding terms in a series like a line of water droplets filling a cup 💧. If each droplet stays a noticeable size forever, the cup will not settle down to a finite total. For a series to possibly converge, the terms must shrink toward $0$. If the terms stay around $5$, or $1$, or even bounce between values like $1$ and $-1$, the sum cannot settle to one finite number.

But here is an important warning: having $\lim_{n\to\infty} a_n = 0$ does not guarantee convergence. It only means the Divergence Test does not rule out convergence. More tests are needed.

How to use the Divergence Test

When you are given a series, students, follow these steps:

Identify the term $a_n$.
Compute $\lim_{n\to\infty} a_n$.
If the limit is not $0$ or does not exist, conclude that $\sum_{n=1}^{\infty} a_n$ diverges.
If the limit is $0$, the test gives no conclusion, so you must try another test.

This makes the Divergence Test a very fast first check. It does not solve every problem, but it can quickly eliminate many impossible cases.

Example 1: constant terms

Consider

$$\sum_{n=1}^{\infty} 4.$$

Here the term is $a_n=4$ for every $n$. Then

$$\lim_{n\to\infty} a_n=4.$$

Since the limit is not $0$, the series diverges by the Divergence Test.

This makes sense because adding $4+4+4+4+\cdots$ keeps growing without bound.

Example 2: terms that do not settle down

Consider

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

The terms are $-1,1,-1,1,\dots$, so the sequence does not approach any single number. Therefore,

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

does not exist.

Because the limit does not exist, the series diverges by the Divergence Test.

Example 3: terms that approach a nonzero number

Consider

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

Rewrite the term as

$$\frac{n^2+1}{n^2}=1+\frac{1}{n^2}.$$

Now compute the limit:

$$\lim_{n\to\infty}\left(1+\frac{1}{n^2}\right)=1.$$

Since the terms do not approach $0$, the series diverges.

What the Divergence Test cannot do

One of the most common mistakes is to think that if $\lim_{n\to\infty} a_n=0$, then the series converges. That is false.

For example, the harmonic series

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

has terms

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

and

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

So the Divergence Test does not show divergence. However, the harmonic series is still divergent. This is a great reminder that the test is only a first filter, not a full decision maker.

Another example is

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

Again,

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

but in this case the series actually converges. So the same term behavior can appear in both convergent and divergent series. The Divergence Test alone cannot tell the difference when the terms go to $0$.

How this fits into Convergence Tests I

In the unit Convergence Tests I, students, you are building a toolkit for deciding whether series converge. The Divergence Test is usually the very first tool because it is quick and easy.

It connects naturally to the other tests:

p-series compare to the form

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

A p-series converges when $p>1$ and diverges when $p\le 1$.

The integral test uses an improper integral to study a series whose terms come from a positive, continuous, decreasing function.

The Divergence Test often comes before these more advanced tests. If a series fails the basic requirement

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

there is no need to apply a more complicated method.

For example, if you see

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

you can simplify the term:

$$\frac{n+2}{n}=1+\frac{2}{n}.$$

Then

$$\lim_{n\to\infty}\left(1+\frac{2}{n}\right)=1,$$

so the series diverges immediately. No integral test or p-series test is needed.

A careful look at the logic

The Divergence Test is based on a necessary condition for convergence. Suppose the series

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

converges. Then its partial sums approach a finite number. For that to happen, the terms must shrink to $0$, because each new term changes the total by less and less.

If the terms do not shrink to $0$, then the total cannot stabilize. This is why the test works.

It is helpful to remember the logic in one sentence:

If the terms do not approach $0$, the series must diverge.

But the reverse is not true:

Even if the terms do approach $0$, the series may still diverge.

That difference is important in Calculus 2 reasoning.

Practice-style examples

Example 4: rational function terms

Consider

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

Divide top and bottom by $n^2$:

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

Then

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

Because the limit is not $0$, the series diverges.

Example 5: terms going to zero but no conclusion

Consider

$$\sum_{n=1}^{\infty} \frac{1}{n+7}.$$

The terms satisfy

$$\lim_{n\to\infty} \frac{1}{n+7}=0.$$

The Divergence Test gives no conclusion. You would need another test, and in fact this series diverges because it behaves like the harmonic series.

Example 6: alternating signs with shrinking size

Consider

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

Then

$$\lim_{n\to\infty} \frac{(-1)^n}{n}=0.$$

Again, the Divergence Test gives no conclusion. The terms do go to $0$, so the test cannot prove divergence. Another test is needed.

Common mistakes to avoid

students, here are some frequent errors students make:

Thinking the Divergence Test proves convergence when the limit is $0$.
Forgetting to evaluate the actual term $a_n$ before applying the test.
Confusing the sequence $\{a_n\}$ with the series $\sum_{n=1}^{\infty} a_n$.
Using the test on a finite sum, which is not an infinite series.
Stopping too early when the limit is $0$ instead of moving to another test.

A good habit is to write the limit clearly before making a conclusion.

Conclusion

The Divergence Test is one of the simplest but most important tools in Convergence Tests I. It says that if

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

or does not exist, then

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

diverges. This test is fast, reliable, and often the first step in analyzing a series. However, if the terms do approach $0$, the test does not tell you whether the series converges or diverges. That is why it belongs to a larger toolkit that also includes p-series and the integral test. students, mastering this test will help you avoid unnecessary work and make smart decisions early in a problem 🔍.

Study Notes

A series has the form $\sum_{n=1}^{\infty} a_n$.
The Divergence Test says: if $\lim_{n\to\infty} a_n \neq 0$ or the limit does not exist, then the series diverges.
If $\lim_{n\to\infty} a_n = 0$, the test gives no conclusion.
The test is a necessary condition for convergence, not a sufficient one.
The harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ has terms going to $0$ but still diverges.
Use the Divergence Test first to quickly eliminate series that cannot converge.
The Divergence Test connects to other Convergence Tests I topics like p-series and the integral test.