# Some fun with special infinite nested radicals

Let us define the following functions:

$$f_n(x)=\sqrt{x^{n}-\sqrt{x^{n+1}- \sqrt{x^{n+2}-\cdots}}}$$ $$g_n(x)=\sqrt{x^{n}+\sqrt{x^{n+1}+ \sqrt{x^{n+2}+\cdots}}}$$

with $$f(x)=f_1(x)$$ and $$g(x)=g_1(x)$$. Very little is known about $$f(x)$$ and $$g(x)$$, except:

• The nested radical converges in both cases if $$x > 1$$
• $$\lim_{x\rightarrow 1^+} f(x) = \frac{-1+\sqrt{5}}{2}$$ and $$g(1) = \frac{1+\sqrt{5}}{2}$$
• $$f(4)=1$$ and $$g(4)=3$$

Let us now focus on the case where $$x>1$$ is an integer. The goal here is to obtain more advanced, interesting results about these nested radicals, maybe even a closed form or some asymptotic formulas.

1. Integer part of the infinite nested radicals

Let $$\lfloor\cdot\rfloor$$ denotes the integer part function. We have:

$$\lfloor f_{2n}(x) \rfloor = x^{n}-\phi(x)\\ \mbox{ } \lfloor g_{2n}(x) \rfloor = x^{n}+\psi(x)$$

with $$\phi(x)=k$$ if $$x\in A_k$$, $$\psi(x)=k$$ if $$x\in B_k$$. The sets $$A_k, B_k$$ are as follows:

• $$A_1=[2,5[, A_2=[5,15[, A_3= [15,33[, A_4=[33,61[, A_5=[61,96[, \cdots$$
• $$B_0=[2,4[, B_1=[4,17[, B_2= [17,38[, B_3=[38,67[, B_4=[67,104[, \cdots$$

2. Fractional part of the infinite nested radicals

Let $$\{\cdot\}$$ denotes the fractional part function. We seem to have:

$$\lim_{n\rightarrow\infty} \{ f_{2n}(x)\}=1-\Big\{\frac{\sqrt{x}}{2}\Big\}\\ \lim_{n\rightarrow\infty} \{ g_{2n}(x)\}=\Big\{\frac{\sqrt{x}}{2}\Big\}$$ Also, the roots of $$\{g_1(x)\}$$ have a very peculiar quadratic distribution. The first few ones, for $$x>0$$, are $$\rho_1=4.0000$$, $$\rho_2=7.3370$$, $$\rho_3=11.6689$$, $$\rho_4=16.9982$$, $$\rho_5=23.3260$$, $$\rho_6=30.6526$$, $$\rho_7=38.9787$$. Furthermore, it seems that

$$\lim_{k\rightarrow\infty} (\rho_{k+2}-2\rho_{k+1} +\rho_k) = 1.$$

Finally, values of $$\{g_1(x)\}$$ for large successive integers $$x$$ lying between two successive roots of $$\{g_1(\cdot)\}$$ tend to be equally spaced as $$x\rightarrow\infty$$. See table below.

3. My question

Actually a few related questions. Feel free to answer the one(s) you are most interested in.

• Many of my results are experimental (thus I often use the word "it seems"). Can you prove some of them?

• Obtain an explicit closed form for all sets $$A_k,B_k$$ used in the definition of $$\phi(x)$$ and $$\psi(x)$$ in section 1. Not sure if it is easy or not.

• We focused on $$n$$ even. What happens for $$n$$ odd? Do we have interesting results? For instance, if $$x=2$$, the successive values of $$\lfloor g_{2n+1}(x)\rfloor$$ are $$2, 3, 6, 12, 23, 45, 91, 181, 362, \cdots$$ (for $$n=0, 1, \cdots$$). I did a reverse lookup on that sequence (see here) but it did not return any result despite the semi-obvious pattern.

• What happens if $$x$$ is not an integer? Any interesting pattern or result?

• Can you derive even more intriguing insights from the empirical results I presented here?

• It looks like the larger $$x$$, the faster my limits are converging. Worth exploring.

• The case $n$ odd can be derived from $n$ even, via the formula $g_{2n+1}(x) = \sqrt{x^{2n+1}+g_{2n+2}(x)}$. Oct 9, 2020 at 15:49
• Yes. I also stated that $f$ converges when $x>1$, however I have no proof for this, though the result sounds easy to prove, albeit a bit more tricky than for $g$. Oct 9, 2020 at 19:07

I am focusing here on $$g(x)$$, with $$x$$ a strictly positive integer. All the results below have been obtained empirically. A proof (or rebuttal) would be welcome. Again, $$\{ \cdot \}$$ represents the floor function.

Let $$b_k=4k^2 + k - 1$$. We have $$B_k=[b_k,b_{k+1}[$$ if $$k>0$$, and $$B_0=[2, 4[$$. Thus we now have a closed form for $$\psi(x)$$ and thus for $$\lfloor g_{2n}(x)\rfloor$$, regardless of $$x$$ and $$n$$, assuming $$x$$ is an integer. In particular, for $$x>0$$, we have:

$$\lfloor g_{2n}(x)\rfloor = x^{n}+\psi(x), \mbox{ with } \psi(x)=\Big\lfloor \small \frac{-1+\sqrt{17+16x}}{8}\Big\rfloor.$$

It also works for $$n=0$$. Let $$\eta(x)=\psi(x) -\lfloor\sqrt{x}/2\rfloor$$. It is equal either to zero (for most $$x$$'s) or one (for $$x=16, 36, 37$$, $$64, 65, 66,\cdots$$). Also, for $$x<16$$, we have the following approximation: $$\{ g_{0}(x)\} \approx \Big\{\frac{\sqrt{x}}{2}\Big\}+\frac{2-\sqrt{5}}{6} (x-4)$$

resulting in

$$g_0(x)=\lfloor g_0(x)\rfloor + \{ g_{0}(x)\} \approx \frac{\sqrt{x}}{2} +\frac{2-\sqrt{5}}{6} (x-4) + 1.$$

The approximation is exact if $$x=1$$ or $$x=4$$. It is also pretty good even if $$x$$ is not an integer. Note that $$g(x)=g_1(x)=g_0^2(x)-1$$.

Another potentially interesting result is this:

$$\lim_{n\rightarrow\infty} \frac{ \{ g_{2n}(x)\}-\{\frac{\sqrt{x}}{2}\} }{ \{ g_{2n+2}(x)\}-\{\frac{\sqrt{x}}{2}\} } = x.$$

Asymptotics and distribution of roots

A better approximation to $$g_0(x)$$, especially for large $$x$$, is the only real, positive solution of the equation $$(y^2−1)^2−y=2x−1$$ with respect to $$x$$. This approximation is also exact for $$x=1$$ and $$x=4$$ and it works for non-integer values of $$x$$. For large $$x$$, we have the following asymptotic expansion for $$g(x)=g_1(x)$$:

$$g(x) =\sqrt{2}\cdot\Big(\sqrt{x}+\frac{1}{8}-\frac{5}{128 \sqrt{x}}+O\Big(\frac{1}{x}\Big)\Big) .$$

The above formula is easy to derive (see Mathematica computation here) and is particularly useful to study the roots of $$\{ g(x)\}$$. If $$x$$ is very large, $$x$$ is a root of $$\{g(x)\}$$ if and only if $$\sqrt{2} (\sqrt{x} + \frac{1}{8})$$ is very close to an integer. Since the first root is $$\rho_1=4$$, an (excellent) approximation to the $$k$$-th root $$\rho_k$$ is the value of $$x$$ satisfying $$\sqrt{2} (\sqrt{x} + \frac{1}{8})=k+2$$. In order words, $$\rho_k = \frac{(k+2)^2}{2} -\frac{k+2}{4\sqrt{2}}+\frac{1}{64} + O\Big(\frac{1}{k}\Big).$$

So, $$g(\rho_k)=k+2$$ and thus $$\{g(\rho_k)\}=0$$ and there is no other root beyond those discussed here. Note that using my approximation, we have $$\rho_1\approx 3.9853$$ while the exact value is $$4$$. The larger $$k$$, the better the approximation since the error term is of the order $$1/k$$ and thus tends to zero as $$k\rightarrow\infty$$.