Square Root of 2 as Sum of Egyptian Fractions

Theorem
The square root of $2$ can be approximated by the following sequence of Egyptian fractions:


 * $\sqrt 2 = 1 + \dfrac 1 3 + \dfrac 1 {253} + \dfrac 1 {218 \, 201} + \dfrac 1 {61 \, 323 \, 543 \, 802} + \cdots$