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Leibniz Criterion — Definition, Formula & Examples

The Leibniz Criterion is another name for the alternating series test. It states that an alternating series converges if its terms decrease in absolute value and approach zero.

Let {bn}\{b_n\} be a sequence of positive real numbers. The series ∑n=1∞(−1)n+1bn\sum_{n=1}^{\infty} (-1)^{n+1} b_n converges if (1) bn+1≤bnb_{n+1} \leq b_n for all nn beyond some index (the sequence is eventually non-increasing), and (2) lim⁡n→∞bn=0\lim_{n \to \infty} b_n = 0. This result is named after Gottfried Wilhelm Leibniz.

Key Formula

∑n=1∞(−1)n+1bn converges if bn+1≤bn and lim⁡n→∞bn=0\sum_{n=1}^{\infty} (-1)^{n+1} b_n \text{ converges if } b_{n+1} \leq b_n \text{ and } \lim_{n \to \infty} b_n = 0
Where:
  • bnb_n = The positive magnitude of the nth term of the series
  • (−1)n+1(-1)^{n+1} = The alternating sign factor

How It Works

To apply the Leibniz Criterion, first confirm the series is alternating — its terms switch between positive and negative. Then isolate the magnitude bnb_n (ignoring the sign). Check that bnb_n is eventually non-increasing, which you can verify by showing bn+1≤bnb_{n+1} \leq b_n or that f(x)f(x) is decreasing for the corresponding continuous function. Finally, verify lim⁡n→∞bn=0\lim_{n \to \infty} b_n = 0. If both conditions hold, the series converges.

Worked Example

Problem: Use the Leibniz Criterion to determine whether the series ∑n=1∞(−1)n+1n\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} converges.
Identify b_n: The magnitude of each term is:
bn=1nb_n = \frac{1}{n}
Check decreasing: Since 1n+1<1n\frac{1}{n+1} < \frac{1}{n} for all n≥1n \geq 1, the sequence bnb_n is strictly decreasing.
bn+1=1n+1≤1n=bn✓b_{n+1} = \frac{1}{n+1} \leq \frac{1}{n} = b_n \quad \checkmark
Check limit: The terms approach zero as nn grows without bound.
lim⁡n→∞1n=0✓\lim_{n \to \infty} \frac{1}{n} = 0 \quad \checkmark
Answer: Both conditions of the Leibniz Criterion are satisfied, so the alternating harmonic series ∑n=1∞(−1)n+1n\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} converges. (It converges to ln⁡2\ln 2.)

Why It Matters

The Leibniz Criterion is one of the first tests students encounter in Calculus II for handling series whose terms change sign. It also reveals the phenomenon of conditional convergence: the alternating harmonic series converges by this criterion even though the ordinary harmonic series diverges.

Common Mistakes

Mistake: Concluding divergence when bnb_n is not decreasing, even though lim⁡bn=0\lim b_n = 0.
Correction: Failure of the decreasing condition means the Leibniz Criterion is inconclusive — it does not prove divergence. You must try another test.