Evaluating the nested radical √1+2√1+3√1+⋯ \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots}}} .

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

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