Evaluating the nested radical $ \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots}}} $. How does one prove the following limit?
$$
  \lim_{n \to \infty}
  \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots \sqrt{1 + (n - 1) \sqrt{1 + n}}}}}
= 3.
$$

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.
