square root of 2

The square root of 2 is an irrational number, the first to have been proved irrational. Its decimal expansion begins 1.41421356237309504880168872420969807856… (sequence http://www.research.att.com/ njas/sequences/A002193A002194 in Sloane’s OEIS) Its simple continued fraction is

$$1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\mathrm{\dots }}}}},$$

periodically repeating the 2. Some call this number Pythagoras’ constant.

There are several different ways to express $\sqrt{2}$ as an infinite product. One way is

$$\sqrt{2}=\prod _{i=0}^{\mathrm{\infty }}\frac{{(4i+2)}^2}{(4i+1)(4i+3)},$$

another is

$$\sqrt{2}=\sum _{i=0}^{\mathrm{\infty }}{(-1)}^{i+1}\frac{(2i-3)!!}{(2i)!!}.$$