Question

What is the value of the sqrt(2+sqrt(2+sqrt(2+...to infinity?

Answer 1

I would say it's approximately equal to square root of 2

Answer 2

\[suppose~ x=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2....}}}}\]
\[Hence~x=\sqrt{2+x},x^2=2+x,x^2-x-2=0,x^2-2x+x-2=0,\]
x(x-2)+1(x-2)=0
(x-2)(x+1)=0
x=2 ,
x=-1 (rejected, because sum of positive quantity cannot be negative.
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