📂Calculus

Alternating Harmonic Series

Definition

The following series is called the alternating harmonic series.

$$\sum\limits_{n = 1}^{\infty} (-1)^{n-1}\dfrac{1}{n} = 1 - \dfrac{1}{2} + \dfrac{1}{3} - \dfrac{1}{4} + \cdots$$

Convergence

The alternating harmonic series converges.

$$\sum\limits_{n = 1}^{\infty} (-1)^{n-1}\dfrac{1}{n} = \ln 2$$

Explanation

On the other hand, the harmonic series diverges.

$$\sum\limits_{n = 1}^{\infty} \dfrac{1}{n} = 1 + \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + \cdots = \infty$$