Square Root of 2 as Sum of Egyptian Fractions

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Theorem

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

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

This sequence is A006487 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof


Sources