Topic 10.3 Notes – The nth Term Test for Divergence

1. What the nth Term Test for Divergence Says

Start with a series:

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

This series is built from the sequence of terms $a_n$. The nth Term Test looks only at those terms.

The Core Fact

If

$$\lim_{n\to\infty} a_n \neq 0 \quad \text{or the limit does not exist,}$$

then

$$\sum a_n \textbf{ diverges.}$$

That’s it.

Why this must be true

A convergent series means the partial sums level off to a finite number. That can only happen if the pieces you’re adding get smaller and smaller, eventually shrinking to 0.

If the terms don’t approach 0, you’re still adding a noticeable amount forever. The total cannot settle down.

Here’s the key logical structure you need to remember:

• Terms do not go to 0 → series diverges
• Terms go to 0 → we don’t know yet

Zero is required for convergence. But zero alone does not guarantee convergence.

2. How to Apply the Test

When you’re handed a series on a quiz or MCQ, this should be your first mental check.

Step 1

Write the limit of the terms:

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

You are not finding the sum. Just the limit of the sequence.

Step 2

Evaluate the limit using standard limit tools:

• Compare highest powers (rational functions)
• Use end behavior rules
• Recognize known limits (like trig, log, exponential behavior)
• Look for horizontal asymptotes

Step 3

Make your conclusion clearly.

• Limit ≠ 0 → diverges
• Limit does not exist → diverges
• Limit = 0 → test is inconclusive

On an FRQ, you must state the reason:

Since $\lim_{n\to\infty} a_n = c \neq 0$, the series diverges by the nth term test.

They want the name of the test.

3. Common Types of Limits You’ll See

Rational Functions in $n$

If

$$a_n = \frac{\text{polynomial}}{\text{polynomial}}$$

Compare degrees:

• Same degree → limit is ratio of leading coefficients → nonzero → diverges
• Numerator degree bigger → limit = ±∞ → diverges
• Denominator degree bigger → limit = 0 → inconclusive

Example idea:

$$a_n = \frac{3n^2 + 1}{5n^2 - 4}$$

Highest powers match → limit = $3/5$ → series diverges immediately.

This shows up constantly on no-calculator multiple choice.

Functions with Known End Behavior

Some expressions level off at a constant that is not zero.

For example:

$$a_n = \arctan(n)$$

The graph makes the end behavior clear:

Graph of $y = \arctan(x)$

As $n \to \infty$, $\arctan(n) \to \frac{\pi}{2}$, which is not zero.

So

$$\sum \arctan(n)$$

diverges.

Other common ones:

• $\ln(n) \to \infty$
• $\sqrt{n} \to \infty$
• $\frac{n}{n+3} \to 1$

If the limit is anything other than 0, you’re done.

Oscillating Sequences

If the terms bounce around and don’t settle at 0, the series diverges.

Example:

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

This alternates between 1 and -1. The limit does not exist → divergence.

But be careful:

$$a_n = \frac{(-1)^n}{n}$$

Now the magnitude shrinks to 0.
The limit equals 0 → nth term test tells you nothing.

This is a classic trap. Students stop too early.

4. What This Test Cannot Do

If

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

you are not finished.

You must move to another test such as:

• Geometric series
• p-series
• Integral test
• Comparison tests
• Alternating Series Test
• Ratio or Root test (coming later)

Classic example:

$$\sum \frac{1}{n}$$

Since $1/n \to 0$, students often say it converges.
It actually diverges.

Terms going to zero is necessary, not sufficient.

5. Fast Exam Mindset

When a series appears:

1. Check $\lim a_n$.
2. If it’s not zero, stop immediately and conclude divergence.
3. If it is zero, switch to a stronger test.

Strong students always do this first. It saves time and prevents overthinking.

On timed MCQs, this can eliminate answers in seconds.