Harmonic series

Example

The harmonic series is the series $$\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots.$$ Its terms tend to $0$, but the series is a divergent series. Its partial sums are the harmonic numbers $H_N = \sum_{n=1}^{N} 1/n$, and they tend to $+\infty$.

Remarks

The node harmonic series diverges gives a proof. It shows that the converse of the term test is false. The partial sums grow very slowly, like $\ln N$. The node harmonic numbers grow like log n makes this precise. The name comes from music. The wavelengths of the overtones of a string are $1/2, 1/3, 1/4, \dots$ of the wavelength of the string.

Explanations

L1 human

The terms $1, 1/2, 1/3, \dots$ become small, but not fast enough. You can group them into blocks that each add up to at least $1/2$, so the total grows without bound.