What is [math]\sqrt {2+\sqrt {2+\sqrt {2+\sqrt {2+...\infty}}}}[/math] ?

The expression (\sqrt {2+\sqrt {2+\sqrt {2+\sqrt {2+...\infty}}}}) represents an infinitely nested radical. To find its value, we can set up an equation based on the recursive nature of the expression.

Let ( x = \sqrt {2+\sqrt {2+\sqrt {2+\sqrt {2+...\infty}}}} ).

Since the expression is infinite and nested, we can substitute ( x ) into itself: [ x = \sqrt {2 + x} ]

Squaring both sides to eliminate the square root: [ x^2 = 2 + x ]

Rearranging this equation into a quadratic form: [ x^2 - x - 2 = 0 ]

Now, solve the quadratic equation: [ (x - 2)(x + 1) = 0 ]

This gives us two possible solutions: [ x = 2 \text{ or } x = -1 ]

However, since the square root of a number is non-negative, we discard the negative solution ( x = -1 ).

Therefore, the value of the infinitely nested radical is: [ x = 2 ]45