Harmonic-Geometric Mean

Let

$$\begin{eqnarray*} \alpha_{n+1}&=&{2\alpha_n\beta_n\over \alpha_n+\beta_n}\\ \beta_{n+1}&=&\sqrt{\alpha_n\beta_n}, \end{eqnarray*}$$

then

$$H(\alpha_0, \beta_0)\equiv \lim_{n\to\infty} a_n = {1\over M({\alpha _0}^{-1},{\beta _0}^{-1})},$$

where $M$ is the Arithmetic-Geometric Mean.

See also Arithmetic Mean, Arithmetic-Geometric Mean, Geometric Mean, Harmonic Mean