Ramanujan's theory

The problem can be solved by Ramanujan's identity. A simplified form is as below (see Nested Functions ). The identity is true if the RHS converges.

$a+b = \sqrt{b^2+a\sqrt{b^2+(a+b)\sqrt{b^2+(a+2b) \sqrt{\cdots}}}}$

Putting $a=2$ and $b=1$ , we have:

$\begin{aligned} 2+1 & = \sqrt{1^2+2\sqrt{1^2+(2+1)\sqrt{1^2+(2+2\cdot 1)\sqrt{1^2 + (2+3\cdot 1) \sqrt{\cdots}}}}} \\ \implies \boxed{3} & = \sqrt{1+2\sqrt{1+3\sqrt{1+4 \sqrt{1+5\sqrt{\cdots}}}}} \end{aligned}$

3=√9

3=√1+8

3=√1+2*4

3=√1+2√16

3=√1+2√1+15

3=√1+2√1+3*5

3=√1+2√1+3√25

3=√1+2√1+3√1+4*6.........