A closed solution for $f^2[n]=1+nf[n+1]$

Question

I attempted to solve $\sqrt{1+\sqrt{1+2\sqrt{1+3\sqrt{\cdots}}}}$ but it leads me to a recursive form $f^2[n]=1+nf[n+1]$.

Is there a closed solution for this difference equation?

Answer 1

From here (Ramanujan's infinite nested radical) we have that $$ S = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}=3. $$

The correct recursive equation and its solution, for arriving at that result (3) is given here in wikipedia.

So your expression is $\sqrt{1+S} = \sqrt{1+3} = 2$.