Background
I was playing around with Ramanujan's infinite root : $$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}}$$ You can write any number as this sequence by limiting the number of square roots in this way; $$ r_{n}(1) = \sqrt{f_{n}(x_{i})}\\\ r_{n}(x) = \sqrt{1+ (x_i - x + 2) \times r_{n}(x-1)} $$ Where:
- $x\rightarrow$ Number of square roots.
- $n\rightarrow$ The number to write.
- $f_{n}(x_i)\rightarrow$ A number that makes $r_{n}(x)=n$ a true statement.
- $x_i\rightarrow$ Initial value for $x$.
For example:
- $r_{3}(2) = \sqrt{1 + 2 \sqrt{16}} = 3$, where $f_{3}(2) = 16$
- $r_{3}(3) = \sqrt{1+ 2 \sqrt{1 + 3 \sqrt{25}}} = 3$, where $f_{3}(3) = 25$
- $r_{4}(3) = \sqrt{1+ 2 \sqrt{1+ 3 \sqrt{\frac{48841}{144}}}} = 4$, where $f_{4}(3) = \frac{48841}{144}$
Problem
My objective is to find a definition for $f$ that doesn't involve recursion, but I haven't been able to do so.
I have been able to define $f$ recursively: $$ f_{n}(1) = n^2\\\ f_{n}(x) = \left(\frac{f_{n}(x-1)-1}{x}\right)^2 $$ Notice how $r_n$ is expanded: $$ r_n(x) = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots\sqrt{1+x\sqrt{f_n(x)}}}}}} $$
Notes
- What I,m trying to do is simplify the definition of $f_n$ not $r_n$.
- $n, x \in N $, and $f_n(x) \in R$.
- I am unsure if this simplification is even possible.
- Ramanujan's infinite root converges to 3$^{[1]}$.
- When $n < 3,\ \lim_{x\to\infty} f_n(x)$ seems to be $0$.
- When $n \geq 3,\ \lim_{x\to\infty} f_n(x)$ seems to be $+\infty$.
- $f_3(x) = (x+2)^2$, provable by induction.
- When $n > 3$, $f$ seems to blow up really fast. (ex: $f_4(10) \approx 2.91159 \times 10^{150}$)
- $f_1(1) = 1$
- $f_1(2) = 0$
- $f_n(3) = \left( \frac{ \left( \frac{ n^2-1 }{2} \right) ^2 - 1}{3} \right) ^2$
- Expansion of $f_n(1) = n^2$
- Expansion of $f_n(2) = \frac{n^4}{4}-\frac{n^2}{2}+\frac{1}{4}$
- Expansion of $f_n(3) = \frac{n^8}{144}-\frac{n^6}{36}-\frac{n^4}{72}+\frac{n^2}{12}+\frac{1}{16}$
- Expansion of $f_n(4) = \frac{n^{16}}{331776}-\frac{n^{14}}{41472}+\frac{n^{12}}{27648}+\frac{5 n^{10}}{41472}-\frac{181 n^8}{165888}+\frac{43 n^6}{13824}+\frac{19 n^4}{9216}-\frac{5 n^2}{512}+\frac{225}{4096}$
- $f_n(10) = \frac{1}{100} \left(1-\frac{1}{81} \left(1-\frac{1}{64} \left(1-\frac{1}{49} \left(1-\frac{1}{36} \left(1-\frac{1}{25} \left(1-\frac{1}{16} \left(1-\frac{1}{144} \left(n^4-2 n^2-3\right)^2\right)^2\right)^2\right)^2\right)^2\right)^2\right)^2\right)^2$
- Some Wolfram code:
(* Function f *)
Subscript[f, n_][1] := n^2
Subscript[f, n_][x_] := ((Subscript[f, n][x - 1] - 1)/x)^2
(* Function r for visualizing *)
Subscript[fView, n_][1] := Defer[n]^2
Subscript[fView, n_][x_] := ((Subscript[fView, n][x - 1] - 1) Defer[x])^Defer[2]
Subscript[r, n_][1, xi_] := Sqrt[Subscript[fView, n][xi]]
Subscript[r, n_][x_, xi_] := Sqrt[1 + (xi - x + 2)*Subscript[r, n][x - 1, xi]]
Subscript[r, n_][x_] := Subscript[r, n][x, x]
Manipulate[{f -> Subscript[f, nn][x], r -> Subscript[r, nn][x], Expand -> Expand[Subscript[f, n][x]]}, {{nn, 1, "n"}, 1, 10, 1}, {x, 1, 10, 1}]