Find the value of
\(\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{2+...}}}}\)
Let x = 1
1 + 1
2 + ....
So we have that
x = 1 / [ 1 + 1/ ( 2 + x) ]
x = 1(2 + x) / (2 + x + 1)
x = (x + 2) / (x + 3)
x (x + 3) = x + 2
x^2 + 3x = x + 2
x^2 + 2x - 2 = 0
Taking the positive solution, x = √3 - 1
Thank you, CPhill.
+99 
