Question

The value of \[2+\sqrt{2+\sqrt{2+\sqrt{2.......}}}\]

Answer 1

gimmick is to put
\[x=2+\sqrt{2+\sqrt{2+\sqrt{2.......}}}\]
\[x-2=\sqrt{2+\sqrt{2+\sqrt{2.......}}}\]
\[(x-2)^2=2+\sqrt{2+\sqrt{2+\sqrt{2.......}}}=x\]

Answer 2

that is solve
\[(x-2)^2=x\]
hard part is proving the limit exists,

Answer 3

I've seen this one before. I think I found it using brute force last time, then someone said something like Limit = 2 + sqrt(Limit), or something like that.