Radicals

OK, so I have just learned about nested radicals, and want to use them as a springboard to meditate on the use of radicals in imagining symmetry in various dimensions. We begin with the infinite nested radicals problem posed by Srivinas Ramanujan.

$? = \sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + \dots}}}$.

His solution involves expressing the geometric series under the radical with the following general formulation

$? = \sqrt{ax + (n + a)^2 + x\sqrt{a(x+n) + (n+a)^2 + (x+n)\sqrt{\dots}}}$.

To solve the equation on the right, we can set it equal to a function of x, $F(x)$

$F(x) = \sqrt{ax + (n + a)^2 + x\sqrt{a(x+n) + (n+a)^2 + (x+n)\sqrt{\dots}}}$

which we can square and simplify to arrive at

$F(x)^2 = ax + (n + a)^2 + xF(x+n)$.

In the documentation on Wikiepdia there was a leap of faith to the further simplified form of

$F(x) = x + n + a$

which I’m still pondering (how do we account for the $F(x + n)$ induction?), but if we believe that, and set $a=0$, $n=1$, and $x=2$, we can simply equate our nested radicals with this $x + n + a$, and demonstrate that the nested radicals to infinity equal $3$. Boom, black box, and all the rest.

For me, the imaginative insight is in expressing the original pattern ($1 + 2\sqrt\dots$) as the polynomial $ax + (n+a)^2 xF(x+n)$, and setting $a=0$. It’s not obvious to me how that was conceived, and I’m interested in finding more documentation that shares the thinking behind this (are there some common paths we can walk to arrive at such subsitutions?).