Harmonic series (mathematics)

In mathematics, the harmonic series is the divergent infinite series:

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots }$

Divergent means that as you add more terms the sum never stops getting bigger. It does not go towards a single finite value.

Infinite means that you can always add another term. There is no final term to the series.

Its name comes from the idea of harmonics in music: the wavelengths of the overtones of a vibrating string are ${\displaystyle {\frac {1}{2}}}$, ${\displaystyle {\frac {1}{3}}}$, ${\displaystyle {\frac {1}{4}}}$, etc., of the string's fundamental wavelength. Apart from the first term, every term of the series is the harmonic mean of the terms either side of it. The phrase harmonic mean also comes from music.