√(2 + √(2 + √(2 + ... )))

What is the value of this infinitely recursive expression?

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.