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.