How prove this Rāmā ujan Aiya kār identity $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+\cdots}}}}}=3$ show that
$$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+\cdots}}}}}=3$$
I know
$$3=\sqrt{1+8}=\sqrt{1+2\sqrt{16}}=\sqrt{1+2\sqrt{1+15}}=\sqrt{1+2\sqrt{1+3\sqrt{1+4\cdot6}}}=\cdots$$

For your question answer is given in this
http://www.isibang.ac.in/~sury/ramanujanday.pdf
