Alternating harmonic series

Example

The alternating harmonic series is $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots.$$ It converges by the alternating series test, since $1/n$ decreases to $0$. Its sum is $\ln 2 \approx 0.6931$.

Remarks

The series of absolute values is the harmonic series, which diverges. So the convergence is conditional. The value $\ln 2$ comes from the Mercator series $\ln(1 + x) = x - x^2/2 + x^3/3 - \cdots$ at $x = 1$, with Abel's theorem. By the Riemann rearrangement theorem, a reordering of the terms can give any real sum.

Explanations

L1 human

The partial sums are $1, 0.5, 0.833, 0.583, 0.783, \dots$. They swing around $\ln 2 \approx 0.693$ and close in on it slowly.