AP Calculus BC

Does the Alternating Harmonic Series Converge?

The sum of negative 1 to the n over n converges, to negative the natural log of 2, about negative 0.6931. It converges CONDITIONALLY: the alternating series test applies, but taking absolute values gives the harmonic series, which diverges.

∑n=1∞(−1)nn\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}

Converges

sum=−ln⁡2\text{sum} = -\ln 2

Settled by the alternating series test, and only conditionally.

The alternating series test

The test needs BOTH conditions on the magnitudes bn=1nb_{n} = \frac{1}{n}, and both hold.

  • bnb_{n} is decreasing: 1n+1<1n\frac{1}{n+1} < \frac{1}{n} for every nn
  • bn→0b_{n} \to 0 as n→∞n \to \infty

With the signs strictly alternating, the partial sums overshoot and undershoot in turn and close in on a single value.

∑n=1∞(−1)nn=−ln⁡2≈−0.6931\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n} = -\ln 2 \approx -0.6931

Why it is only conditional

Take absolute values and the alternation disappears, leaving the harmonic series, which diverges. Converging as written while failing in absolute value is exactly the definition of conditional convergence.

∑n=1∞∣(−1)nn∣=∑n=1∞1n=∞\sum_{n=1}^{\infty}\left|\frac{(-1)^{n}}{n}\right| = \sum_{n=1}^{\infty}\frac{1}{n} = \infty

Conditional convergence is fragile

Rearranging the terms of a conditionally convergent series can change its sum to ANY value you like. That cannot happen to an absolutely convergent series, which is the real reason the distinction is worth making.

The mistakes students make

  • Calling it absolutely convergent because it converges. Absolute convergence requires the series of absolute values to converge, and here it does not.
  • Checking only that the terms go to 00. The magnitudes must ALSO be decreasing, and a series can fail on that condition alone.
  • Getting the sign wrong. Starting at n=1n = 1 the first term is −1-1, so the sum is −ln⁡2-\ln 2; the version with (−1)n+1(-1)^{n+1} sums to +ln⁡2+\ln 2.

Not sure which test a series wants?

The Convergence Test Chooser walks the decision in order: nth term first, then geometric and p-series pattern matching, then alternating structure, then the ratio test, and finally the comparison family.

Frequently asked questions

Does the alternating harmonic series converge?

Yes, conditionally, to −ln⁡2-\ln 2 when written with (−1)n(-1)^{n} from n=1n = 1.

Why conditionally and not absolutely?

Because the absolute values give the harmonic series, which diverges.

What does the sign convention change?

(−1)n(-1)^{n} from n=1n = 1 starts negative and sums to −ln⁡2-\ln 2; (−1)n+1(-1)^{n+1} starts positive and sums to +ln⁡2+\ln 2.