Alternating Series and Leibniz Test: Full Explanation

An alternating series is a series whose terms switch sign at every step, written as Σ (−1)n bn or Σ (−1)n+1 bn with bn ≥ 0. The Leibniz test (also called the alternating series test) says that such a series converges whenever the sizes bn decrease steadily to zero. It is the single most-used test for series with alternating signs.

This test is important because alternating series show up constantly: the Taylor series for ln(1+x), arctan x and sin x all alternate, and error estimates in numerical approximations rely on them. Without the Leibniz test, proving that these familiar series converge would be far harder.

Key Concepts

• Leibniz test: if bn ≥ 0, bn+1 ≤ bn for all n (decreasing), and bn → 0, then Σ (−1)n bn converges.
• Partial sum behaviour: the even partial sums S2N increase and the odd partial sums S2N+1 decrease, trapping the true sum S between them like a closing pair of jaws.
• Remainder estimate: the error after N terms satisfies |RN| = |S − SN| ≤ bN+1, the size of the first omitted term.
• Alternating harmonic series: Σ (−1)n+1/n converges to ln 2 ≈ 0.6931 by the Leibniz test, since 1/n decreases to 0.
• Decreasing is essential: terms that merely tend to zero without decreasing need not give a convergent alternating series.

Formulas

• Leibniz test conditions: bn ≥ 0, bn+1 ≤ bn, lim bn = 0 ⇒ Σ (−1)n bn converges.
• Error bound: |S − SN| ≤ bN+1. To guarantee an error below ε, choose N with bN+1 < ε.
• Terms needed: for the alternating harmonic series, bN+1 = 1/(N+1), so an error below 0.01 needs N ≥ 99.
• Bracket property: S2N ≤ S ≤ S2N+1, so two consecutive partial sums always bracket the true value.

Worked Example

Approximate the alternating harmonic series S = Σn=1∞ (−1)n+1/n using four terms, and bound the error.

1. Check the conditions: bn = 1/n is positive, decreasing, and tends to 0, so the Leibniz test applies and the series converges.
2. Compute S4 = 1 − 1/2 + 1/3 − 1/4. With decimals: 1 − 0.5 + 0.3333 − 0.25 = 0.5833.
3. The error bound is the next term: |S − S4| ≤ b5 = 1/5 = 0.2.
4. The true value is ln 2 ≈ 0.6931, so the actual error is |0.6931 − 0.5833| = 0.1098, comfortably inside the guaranteed bound of 0.2.

Common Misconceptions

• "The error equals the next term." The next term is only an upper bound on the error, not the error itself; the true error is usually smaller.
• "Any series with terms going to zero and alternating signs converges." The sizes must also be decreasing; without monotonicity the test does not apply.
• "The Leibniz test also shows absolute convergence." It never does. The alternating harmonic series converges by Leibniz but Σ 1/n diverges, so the convergence is conditional.

Real-World Applications

• Estimating ln 2, π/4 (from the arctan series) and other constants to guaranteed accuracy using the simple error bound.
• Bounding truncation error when software evaluates alternating Taylor series for sin x, cos x and e−x.
• Proving convergence of alternating Fourier series used in electrical engineering signal analysis.

The Leibniz test needs only three things: non-negative terms, steadily decreasing sizes, and a limit of zero. When those hold, the series converges, and the first dropped term hands you a free error bound. Just remember that it proves convergence only, never absolute convergence, and that the decreasing condition cannot be skipped.