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Alternating Harmonic Series — Definition, Formula & Examples

The alternating harmonic series is the infinite series 1−12+13−14+⋯1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots, formed by giving the terms of the harmonic series alternating positive and negative signs. It converges to ln⁡2\ln 2, even though the ordinary harmonic series diverges.

The alternating harmonic series is defined as ∑n=1∞(−1)n+1n\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}. By the alternating series test, it converges, and its sum equals ln⁡2\ln 2. Because the corresponding series of absolute values ∑1/n\sum 1/n diverges, the alternating harmonic series converges conditionally but not absolutely.

Key Formula

∑n=1∞(−1)n+1n=1−12+13−14+⋯=ln⁡2\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots = \ln 2
Where:
  • nn = Positive integer index of summation, starting at 1
  • (−1)n+1(-1)^{n+1} = Produces the alternating sign pattern: +, −, +, −, …

Worked Example

Problem: Use the first four partial sums of the alternating harmonic series to observe how it approaches ln 2 ≈ 0.6931.
S₁: Take just the first term.
S1=1=1.0000S_1 = 1 = 1.0000
S₂: Subtract the second term.
S2=1−12=0.5000S_2 = 1 - \frac{1}{2} = 0.5000
S₃: Add the third term.
S3=1−12+13≈0.8333S_3 = 1 - \frac{1}{2} + \frac{1}{3} \approx 0.8333
S₄: Subtract the fourth term.
S4=1−12+13−14≈0.5833S_4 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} \approx 0.5833
Answer: The partial sums oscillate above and below the true value ln⁡2≈0.6931\ln 2 \approx 0.6931, gradually closing in on it. This oscillation is characteristic of alternating series.

Visualization

Why It Matters

The alternating harmonic series is the textbook example of conditional convergence — it converges, yet rearranging its terms can make it sum to any real number (by the Riemann rearrangement theorem). Understanding this series solidifies the distinction between absolute and conditional convergence, a key topic on the AP Calculus BC exam and in university analysis courses.

Common Mistakes

Mistake: Assuming convergence of the alternating harmonic series implies the harmonic series also converges.
Correction: The harmonic series ∑1/n\sum 1/n diverges. The alternating signs are essential for convergence here. This is exactly what makes the series conditionally convergent rather than absolutely convergent.