📂Calculus

Harmonic Series

Definition

The following series is called the harmonic series.

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

Explanation

It is a representative counterexample to the divergence test. That is, the harmonic sequence converges, but the harmonic series diverges.

$$\lim\limits_{n \to \infty} \dfrac{1}{n} = 0 \quad \text{ but } \quad \sum\limits_{n = 1}^{\infty} \dfrac{1}{n} = \infty$$

On the other hand, the alternating harmonic series converges.

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

Convergence

The harmonic series diverges.

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