How does one prove the following limit?

lim

**Answer**

This is Ramanujan’s famous nested radical.

More information can be found here: http://www.isibang.ac.in/~sury/ramanujanday.pdf

See Also: http://mathworld.wolfram.com/NestedRadical.html (number 26).

Apparently, this is how he came up with it (sorry, no reference for this claim).

Start with

3 = \sqrt{9} = \sqrt{1 + 8} = \sqrt{1 + 2 \cdot 4}

= \sqrt{1 + 2\sqrt{16}} = \sqrt{1 + 2\sqrt{1 + 3 \cdot 5}}

= \sqrt{1 + 2\sqrt{1 + 3 \sqrt{25}}} = \sqrt{1 + 2\sqrt{1 + 3 \sqrt{1 + 4 \cdot 6}}}

etc.

**Attribution***Source : Link , Question Author : Community , Answer Author : Aryabhata*