$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+...}}}}}$
My Attempt: I tried to use the regular way.
$A=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+...}}}}}$
$A^2=1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+...}}}}$
But then I saw that nothing happened twice and I stopped. Any hints?
It's better to don't use limits but if you used no problem. If there is an answer that doesn't use limits will accept.
update1: The question that look likes this wants to find the limit but I want a much easier way to solve it. But if there isn't any answer easier answer the limition one will accept.
update2: You solve the problem when you know the answer is 3 think that you don't know that the answer is 3 then solve.