Wednesday, April 1, 2009

√(2 + √(2 + √(2 + ... )))

What is the value of this infinitely recursive expression?
root (2+ root(2 + root(2 + ...)))

The kids all guessed it is two by successive approxmation:

√2 = 1.414
√(2 + √2) = √ 3.414 = 1.874
√(2 + √(2 + √2)) = √3.874 = 1.9615

Algebraically we can write set x= √(2 + √(2 + √(2 + ...))). Then we observe that the inner expression is also x, so we have x=√(2+x).

Solving for x we get
x2 = 2 + x,
or
x2 - x -2 = 0.

You could use the quadratic equation or you could guess the factorization. Somehow, by hook or by crook you get
(x-2)(x+1)=0.
And so x=2, as the kids guessed. The kids observed that it's the same trick for solving a geometric series:
1+1/2+1/4+1/8+ ... = x,
so
1 + x/2 = x,
which has solution x=2.

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I'm research faculty at MIT, and Chief Architect at Tokutek.