How to use the nth term test for divergence

What You Need to Know

The big idea (why this matters)

Before you try any “real” convergence test (ratio, integral, comparison, etc.), you should always check the nth term test for divergence. It’s the fastest way to prove a series cannot converge.

The theorem (necessary condition for convergence)

For an infinite series $\sum_{n=1}^{\infty} a_n$:

• If $\sum_{n=1}^{\infty} a_n$ converges, then the terms must satisfy
  $\lim_{n\to\infty} a_n = 0.$

Contrapositive (the actual “test” you use):

• If $\lim_{n\to\infty} a_n \neq 0$ or the limit does not exist (DNE), then
  $\sum_{n=1}^{\infty} a_n \text{ diverges.}$

Critical reminder: This test can only prove divergence. If $\lim_{n\to\infty} a_n = 0,$ the test is inconclusive (the series might converge or diverge).

When you use it

Use it immediately when you’re given a series. It’s especially useful when:

• The terms clearly approach a nonzero constant (like $\frac{n}{n+1}$).
• The terms oscillate and don’t settle to a limit (like $(-1)^n$).
• The expression is complicated, but the limit is easy (dominant term thinking).

Step-by-Step Breakdown

How to apply the nth term test (every time)

1. Identify the general term $a_n$.

   • Your series must be written as $\sum a_n$.
   • Ignore the sigma for the moment; focus on the expression being summed.

2. Compute the limit $\lim_{n\to\infty} a_n$.

   • Use algebra (dominant terms), known limits, or squeeze as needed.

3. Decide using the limit:

   • If the limit is nonzero: series diverges.
   • If the limit DNE: series diverges.
   • If the limit is zero: test inconclusive → move on to another test.

Mini-annotated walkthroughs

Walkthrough A: limit is nonzero → immediate divergence

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

• Here $a_n = \frac{n}{n+1}$
• Compute limit:
  $\lim_{n\to\infty} \frac{n}{n+1} = \lim_{n\to\infty} \frac{1}{1+\frac{1}{n}} = 1$
• Since $\lim a_n = 1 \neq 0$, the series diverges.

Walkthrough B: limit is zero → inconclusive (do NOT claim convergence)

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

• $a_n = \frac{1}{n}$
• $\lim_{n\to\infty} \frac{1}{n} = 0$
• nth term test says nothing else. (This series diverges, but for a different reason.)

Walkthrough C: limit does not exist → divergence

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

• $a_n = (-1)^n$
• The sequence $(-1)^n$ oscillates between $-1$ and $1$, so
  $\lim_{n\to\infty} (-1)^n \text{ DNE}$
• Therefore the series diverges.

Key Formulas, Rules & Facts

Core rule table

Practical limit tools you’ll use a lot

What “diverges by nth term test” actually means

To say “diverges by nth term test,” you must show:

• Either $\lim_{n\to\infty} a_n \neq 0$
• Or $\lim_{n\to\infty} a_n$ does not exist

Then conclude:
$\sum_{n=1}^{\infty} a_n \text{ diverges.}$

You are not allowed to conclude “converges” from $\lim a_n = 0$.

Examples & Applications

Example 1: Constant-ish terms (most common quick kill)

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

• $a_n = \frac{3n+1}{n-2}$
• Divide by $n$:
  $\lim_{n\to\infty} \frac{3+\frac{1}{n}}{1-\frac{2}{n}} = \frac{3}{1} = 3$
• Since $3 \neq 0$, the series diverges by nth term test.

Example 2: Oscillation (limit DNE)

Determine if $\sum_{n=1}^{\infty} \cos(n\pi)$ converges.

• $a_n = \cos(n\pi) = (-1)^n$
• Limit does not exist (alternates between $1$ and $-1$)
• Therefore the series diverges by nth term test.

Example 3: A “sneaky” zero limit (inconclusive)

Determine what nth term test says about $\sum_{n=1}^{\infty} \frac{\sin(n)}{n}$.

• $a_n = \frac{\sin(n)}{n}$
• Since $-1 \le \sin(n) \le 1$, you have
  $-\frac{1}{n} \le \frac{\sin(n)}{n} \le \frac{1}{n}$
• Both bounds go to $0$, so by squeeze:
  $\lim_{n\to\infty} \frac{\sin(n)}{n} = 0$
• nth term test is inconclusive (you need a different convergence test).

Example 4: Trap series where terms go to zero but it still diverges

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

• $a_n = \frac{1+(-1)^n}{n}$
• For odd $n$, $1+(-1)^n=0$ so $a_n=0$.
• For even $n$, $1+(-1)^n=2$ so $a_n=\frac{2}{n}$.
• Limit:
  $\lim_{n\to\infty} a_n = 0$ (because along even terms $\frac{2}{n}\to 0$ and odd terms are $0$)
• nth term test: inconclusive.

But notice the even-term subseries:
$\sum_{k=1}^{\infty} a_{2k} = \sum_{k=1}^{\infty} \frac{2}{2k} = \sum_{k=1}^{\infty} \frac{1}{k}$
which diverges, so the original diverges too.

Exam takeaway: $\lim a_n = 0$ does not guarantee anything.

Common Mistakes & Traps

1. Mistake: Concluding convergence when $\lim_{n\to\infty} a_n = 0$

   • What goes wrong: You treat the condition as “if and only if.”
   • Why wrong: $\lim a_n = 0$ is necessary but not sufficient.
   • Fix: Train yourself to say “inconclusive” automatically when the limit is $0$.

2. Mistake: Using the test on the wrong expression (not the nth term)

   • What goes wrong: You might take the limit of the partial sum $S_n$ or something else.
   • Why wrong: The test is about $a_n$ (the term), not $S_n$.
   • Fix: Always rewrite as $\sum a_n$ and explicitly label $a_n=$.

3. Mistake: Forgetting that “limit DNE” still forces divergence

   • What goes wrong: You only check whether the limit is nonzero, not whether it exists.
   • Why wrong: If $a_n$ oscillates, the terms do not approach $0$.
   • Fix: If you see $(-1)^n$, trig at integers, or piecewise behavior, consider DNE.

4. Mistake: Algebra errors in rational limits

   • What goes wrong: You mis-handle dividing by powers of $n$.
   • Why wrong: One small algebra slip flips the conclusion.
   • Fix: For $\frac{p(n)}{q(n)}$, compare degrees or divide numerator and denominator by the highest power of $n$ in the denominator.

5. Mistake: Thinking $a_n \to \infty$ is a special case

   • What goes wrong: You hesitate because the rule mentions “not zero.”
   • Why wrong: If $a_n \to \infty$, then definitely $\lim a_n \neq 0$.
   • Fix: Any nonzero or infinite limit means divergence.

6. Mistake: Confusing sequence convergence with series convergence

   • What goes wrong: You mix up “$a_n$ converges” with “$\sum a_n$ converges.”
   • Why wrong: A convergent sequence can still have a divergent series (classic: $a_n=\frac{1}{n}$).
   • Fix: Remember: series convergence is about partial sums, and nth term test is just a quick necessary check.

7. Mistake: Applying it to a shifted index incorrectly

   • What goes wrong: You worry if the series starts at $n=0$ or $n=5$.
   • Why wrong: Finite starting index shifts don’t affect the limit.
   • Fix: The limit as $n\to\infty$ ignores the first finitely many terms.

Memory Aids & Quick Tricks

Quick Review Checklist

• [ ] You rewrote the series as $\sum a_n$ and clearly identified $a_n$.
• [ ] You computed $\lim_{n\to\infty} a_n$ (or showed it DNE).
• [ ] If $\lim a_n \neq 0$ (including $\pm\infty$): you concluded diverges.
• [ ] If $\lim a_n$ DNE: you concluded diverges.
• [ ] If $\lim a_n = 0$: you wrote inconclusive (and didn’t stop there).
• [ ] You didn’t confuse $a_n$ with partial sums $S_n$.
• [ ] You remembered: nth term test never proves convergence.

One last push: check the term limit first—free points when it works.