Silver Ratio

The quantity defined by the Continued Fraction

$$\delta_S \equiv [2,2,2,\ldots]=2+{1\over 2+{\strut\displayst... ... 2+{\strut\displaystyle 1\over\strut\displaystyle 2+\cdots}}}.$$

It follows that

$$(\delta_S-1)^2 = 2,$$

so

$$\delta_S = \sqrt{2}+1=2.41421\ldots.$$

See also Golden Ratio, Golden Ratio Conjugate