What the nth term test is and how to apply it
The nth term test, also called the Test for Divergence, checks whether a series Σ aₙ can converge by examining the limit of its terms. If lim aₙ ≠ 0 as n → ∞, the series must diverge. If the limit is 0, the test is inconclusive; the series may converge or diverge, and stronger tests are required. This evergreen explainer defines the test, shows how to compute the limit, provides examples, clarifies common misuses, and explains how it fits into a broader convergence-testing workflow.
Definition and statement in formal terms
For an infinite series Σ aₙ (with terms indexed starting at some integer n = N), the nth term test states: if lim_{n→∞} aₙ does not exist or exists and is nonzero, then Σ aₙ diverges. The contrapositive is logically certain: if Σ aₙ converges, then lim_{n→∞} aₙ = 0. The converse is not guaranteed; a limit of zero does not prove convergence. This makes the test useful only for detecting divergence, not confirming convergence.
Formal condition and conclusion
- If lim_{n→∞} aₙ ≠ 0 (or the limit does not exist): series diverges.
- If lim_{n→∞} aₙ = 0: test is inconclusive; use other tests.
Step-by-step procedure to use the test
Follow these steps to apply the nth term test correctly:
- Identify the general term aₙ of the series.
- Compute the limit L = lim_{n→∞} aₙ, using standard limit techniques (algebra, L’Hôpital’s rule, known limits, or asymptotic comparisons).
- Interpret the result:
- If L ≠ 0 or L does not exist, the series diverges.
- If L = 0, the test is inconclusive; proceed to other convergence tests (comparison, ratio, root, integral, etc.).
Practical tips for computing the limit
- Simplify the expression algebraically before taking the limit.
- For rational functions of n, compare degrees: if numerator degree
- For exponential, factorial, or logarithmic terms, use growth-rate hierarchy: factorial and exponential (with base >1) dominate polynomials and logarithms.
Worked examples and counterexamples
Examples clarify when the test confirms divergence and when it yields no conclusion.
Example 1: Divergence confirmed
Consider Σ (2n + 1)/(n + 3). The term aₙ = (2n + 1)/(n + 3) tends to 2 as n → ∞. Since L ≠ 0, the nth term test confirms divergence.
Example 2: Inconclusive (limit zero)
Consider the harmonic series Σ 1/n. Here aₙ = 1/n and lim 1/n = 0, yet the series diverges. This illustrates why limit zero does not guarantee convergence.
Example 3: Divergence with oscillating terms
For Σ (−1)^n, the terms alternate between −1 and 1, so the limit does not exist. By the nth term test, the series diverges.
Common misuses and limitations
Misapplying the nth term test is a common error. Since a zero limit is inconclusive, using the test alone and concluding convergence can lead to false results. The test cannot handle series with positive terms that decay slowly (like the harmonic series) and it does not apply to improper integrals or alternating series behavior beyond divergence detection. It is a quick screen for divergence only.
How the nth term test fits into testing strategies
Effective convergence analysis uses the nth term test as an early screen, followed by more powerful methods as needed.
| Test | Best used when | Outcome types |
|---|---|---|
| nth term (divergence) test | Quick check of term behavior at infinity | Divergence confirmed; inconclusive if limit 0 |
| Geometric series test | Series of the form Σ arⁿ | Converges if |r| |
| p-series test | Series of the form Σ 1/nᵖ | Converges if p > 1; diverges if p ≤ 1 |
| Comparison test (direct) | Terms resemble a known benchmark series | Convergence or divergence by comparison |
| Ratio test | Terms involve factorials, exponentials, or powers | Absolute convergence, divergence, or inconclusive |
| Root test | Terms involve nth powers | Absolute convergence, divergence, or inconclusive |
Decision flow for series convergence
- Step 1: Apply the nth term test. If it diverges, stop; the series diverges.
- Step 2: If limit is 0, identify series type (geometric, p-series, rational, exponential, factorial).
- Step 3: Choose an appropriate confirmatory test (ratio, root, comparison, integral, alternating series, etc.).
- Step 4: Interpret results carefully and, if needed, combine multiple tests.