Ramanujan's Infinite Nested Roots

Theorem

 * $3 = \sqrt {1 + 2 \sqrt {1 + 3 \sqrt { 1 + \cdots} } }$

Proof
We have:

In order for this sequence to continue, it needs to be shown that:


 * $n + 1 = \sqrt {1 + n \left({n + 2}\right)}$

Thus:

The result follows.