How to use the nth term test for divergence

What You Need to Know

The big idea (why this matters)

When you’re given an infinite series $\sum_{n=1}^{\infty} a_n$, the nth term test for divergence (also called the divergence test or term test) is the fastest “sanity check” you should try first.

It tells you a necessary condition for convergence:

• If a series converges, then its terms must approach $0$.
• If the terms do not approach $0$ (or the limit doesn’t exist), the series definitely diverges.

Warning: This test can only prove divergence. If the limit is $0$, you learn nothing about convergence.

Core theorem (state it precisely)

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

• If $\lim_{n\to\infty} a_n \neq 0$, then $\sum_{n=1}^{\infty} a_n$ diverges.
• If $\lim_{n\to\infty} a_n$ **does not exist**, then $\sum_{n=1}^{\infty} a_n$ diverges.
• If $\lim_{n\to\infty} a_n = 0$, the test is inconclusive (series may converge or diverge).

When you use it

Use it:

• Immediately when you see a series (it’s quick).
• To catch “trick” series where terms don’t go to zero (common on AP Calc).
• As a first decision point before choosing other tests (geometric, p-series, comparison, ratio, alternating, etc.).

Step-by-Step Breakdown

Procedure (do this every time)

1. Identify the term $a_n$ in the series $\sum a_n$.
2. Compute $\lim_{n\to\infty} a_n$.
   • Use algebra simplification first.
   • If it’s a rational expression in $n$, divide by the highest power of $n$.
   • If it’s exponential/factorial, recall growth rates (factorials/exponentials dominate polynomials).
3. Decide:
   • If the limit is not $0$ (including $\pm\infty$): diverges.
   • If the limit does not exist (oscillates): diverges.
   • If the limit is exactly $0$: inconclusive → switch to another convergence test.

Mini worked walkthroughs (annotated)

Example A: Limit is not zero → diverges

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

1. $a_n = \frac{n}{n+1}$
2. $\lim_{n\to\infty} \frac{n}{n+1} = \lim_{n\to\infty} \frac{1}{1+\frac{1}{n}} = 1$
3. Since the limit $\neq 0$, the series diverges.

Example B: Limit does not exist (oscillates) → diverges

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

1. $a_n = (-1)^n$
2. $\lim_{n\to\infty} (-1)^n$ does not exist (it flips between $-1$ and $1$).
3. So the series diverges.

Example C: Limit is zero → inconclusive

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

1. $a_n = \frac{1}{n}$
2. $\lim_{n\to\infty} \frac{1}{n} = 0$
3. Test is inconclusive (this series actually diverges, but you need a different test).

Decision point: $\lim a_n = 0$ means “it might converge,” not “it does converge.”

Key Formulas, Rules & Facts

The test in one table

Necessary condition for convergence (the “must” rule)

If $\sum_{n=1}^{\infty} a_n$ converges, then:

• $\lim_{n\to\infty} a_n = 0$.

The contrapositive is what you use for the nth term test:

• If $\lim_{n\to\infty} a_n \neq 0$ or DNE, then $\sum a_n$ diverges.

Quick limit evaluation facts that show up a lot

• Rational functions: for $a_n = \frac{p(n)}{q(n)}$, compare degrees.

  • If degree numerator = degree denominator, limit is ratio of leading coefficients.
  • If degree numerator < degree denominator, limit $=0$.
  • If degree numerator > degree denominator, limit is $\pm\infty$ (or DNE by sign).

• Growth rates (useful for deciding limits quickly):

  • Factorials dominate exponentials dominate powers dominate logs, i.e.
    $n! \gg c^n \gg n^p \gg \ln(n)$
    So examples:
  • $\lim_{n\to\infty} \frac{n^5}{2^n} = 0$
  • $\lim_{n\to\infty} \frac{2^n}{n!} = 0$

• Oscillation warning: if $a_n$ contains $(-1)^n$, $\sin(n)$, $\cos(n)$, etc., the limit might **not exist** unless there’s a “shrinking factor” like $\frac{1}{n}$.

  • $\lim_{n\to\infty} \frac{\sin(n)}{n} = 0$ (inconclusive for the series)
  • $\lim_{n\to\infty} \sin(n)$ DNE (so the series diverges)

Examples & Applications

1) Classic “looks like it might converge” but fails immediately

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

• $a_n = \frac{3n+1}{n} = 3 + \frac{1}{n}$
• $\lim_{n\to\infty} a_n = 3$
• Since the limit $\neq 0$, the series diverges.

Exam vibe: They want you to notice you don’t even need a “real” convergence test.

2) Alternating-looking series that still diverges (because terms don’t go to 0)

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

• $a_n = (-1)^n \frac{n}{n+1}$
• Magnitude approaches $1$, and the sign flips.
• $\lim_{n\to\infty} a_n$ does not exist (oscillates between values near $-1$ and $1$).
• Therefore the series diverges by nth term test.

Key insight: “Alternating” does not automatically mean convergent. You still need $a_n \to 0$.

3) Limit is zero, but series diverges anyway (test is inconclusive)

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

• $\lim_{n\to\infty} \frac{1}{n} = 0$
• nth term test says: inconclusive.
• Reality: diverges (harmonic series).

Exam vibe: They’re testing whether you falsely conclude convergence from $a_n \to 0$.

4) Limit is zero, and series converges (still inconclusive)

$\sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^n$

• $a_n = \left(\frac{1}{2}\right)^n$
• $\lim_{n\to\infty} a_n = 0$
• nth term test: inconclusive.
• Reality: converges (geometric series).

Takeaway: Same nth-term result (limit $0$) can correspond to both convergence and divergence.

5) A “hidden” nonzero limit inside a rational expression

$\sum_{n=1}^{\infty} \frac{n^2 + 7}{2n^2 - 1}$

• $a_n = \frac{n^2 + 7}{2n^2 - 1}$
• Divide by $n^2$: $a_n = \frac{1 + \frac{7}{n^2}}{2 - \frac{1}{n^2}}$
• $\lim_{n\to\infty} a_n = \frac{1}{2}$
• Since the limit $\neq 0$, the series diverges.

Exam vibe: Quick algebra → instant divergence.

Common Mistakes & Traps

1. Mistake: Thinking $\lim_{n\to\infty} a_n = 0$ means the series converges

   • What goes wrong: You confuse a necessary condition with a sufficient one.
   • Why wrong: Many divergent series still have terms that go to $0$ (harmonic, p-series with $p \le 1$).
   • Fix: Train your brain to say: “Limit $0$ → inconclusive.”

2. Mistake: Applying the test to the wrong thing (partial sums instead of terms)

   • What goes wrong: You compute $\lim_{n\to\infty} S_n$ where $S_n = \sum_{k=1}^n a_k$.
   • Why wrong: The nth term test is about $a_n$, not $S_n$.
   • Fix: Always write “$a_n = \dots$” first.

3. Mistake: Forgetting that “limit does not exist” also forces divergence

   • What goes wrong: You only look for a nonzero limit but ignore oscillation.
   • Why wrong: If $a_n$ doesn’t settle to a single number, it definitely doesn’t go to $0$ in the needed way.
   • Fix: If terms flip or wander (like $(-1)^n$), explicitly say “limit DNE → diverges.”

4. Mistake: Concluding convergence from $a_n$ becoming small “quickly”

   • What goes wrong: You see something like $\frac{1}{\sqrt{n}}$ or $\frac{\ln(n)}{n}$ and assume convergence.
   • Why wrong: Small terms are not enough; you need an actual convergence test.
   • Fix: Use nth term test only as a gatekeeper. If it passes (limit $0$), move on.

5. Mistake: Not simplifying before taking the limit

   • What goes wrong: You try to evaluate $\lim \frac{n}{n+1}$ without dividing by $n$, or you mis-handle leading terms.
   • Why wrong: Algebra is the difference between seeing $1$ instantly vs. getting stuck.
   • Fix: For rational expressions, divide numerator/denominator by the highest power of $n$.

6. Mistake: Misusing L’Hôpital’s Rule

   • What goes wrong: You apply L’Hôpital’s Rule automatically to sequences.
   • Why wrong: L’Hôpital is for functions with indeterminate forms; for sequences you can sometimes treat $n$ as continuous, but that’s beyond what you typically need here.
   • Fix: Stick to algebraic limit tricks and growth-rate reasoning unless your teacher explicitly allows otherwise.

7. Mistake: Ignoring domain issues and undefined terms

   • What goes wrong: You assume $a_n$ is defined for all $n \ge 1$, but something like $\ln(n-3)$ starts later.
   • Why wrong: A series must be well-defined from some index onward.
   • Fix: If needed, shift the start index (finite number of initial terms doesn’t affect convergence), then apply the test.

Memory Aids & Quick Tricks

Quick self-check phrase: “I’m testing the terms, not the sum.”

Quick Review Checklist

• You’re given $\sum a_n$ → first ask: what is $\lim_{n\to\infty} a_n$?
• If $\lim a_n \neq 0$, is $\pm\infty$, or DNE → series diverges.
• If $\lim a_n = 0$ → inconclusive (pick another test).
• Watch for oscillation: $(-1)^n$, $\sin(n)$, $\cos(n)$ can make limits DNE.
• For rational expressions in $n$: divide by the highest power of $n$.
• Don’t confuse $a_n$ with partial sums $S_n$.
• Remember: Every convergent series has $a_n \to 0$, but not every series with $a_n \to 0$ converges.

You’ve got this—use the nth term test as your fast first filter, then move to a stronger test when it’s inconclusive.