Find the value of
$$\sqrt{1+2\sqrt{1+3\sqrt{1+....}}}$$
I have done a similar question before in which only a single number is involved, for example $\sqrt{2+2\sqrt{2+2\sqrt{2+....}}}$ which can easily be solved due to repetition of the expression. For this question, when I set the whole thing equal to $f(x)$, I got this
$$[f(x)]^2=xf(x+1)+1$$
After that, I'm not able to solve this functional equation further. Is there any particular method for solving these types of functional equations?
Any help will be appreciated.
Thanks!
