Yet another nested radical
Consider $$F(x) = \sqrt{x -\sqrt{2x - \sqrt{3x - \cdots}}}$$
I believe I can prove (with some handwaving) that
- $F$ does converge everywhere in $\mathbb{C}$
- $\Im F = 0$ for sufficiently large real $x$ (actually larger than $x0 \approx 0.5243601\dots$ Does this number ring a bell?)
- Coincidentally $F(x0) = 0$
Weird things happen in the limit to $0$. Obviously, $F(0) = 0$. However, it seems that $$\lim_{x \to +0}F(x) = \overline{\zeta} $$ $$\lim_{x \to -0}F(x) = \zeta $$ where $\zeta = \frac{1 + i\sqrt{3}}{2}$ is a usual cubic root of $-1$. Moreover, $F$ seems to reach one of those as $x$ approaches $0$ at a rational angle. I understand that this may well be a computational artifact (still making no sense to me), but proving or refuting these limits is definitely out of my league.
Any help?
Solution 1:
This is not an answer, but some data for illustration. There seem to be critical values $n_x$ for some $x$ near zero, such that the partial evaluations becomes calm from initially complex to finally real values. I've interpreted your function for some given n as $$f(x,n)=\sqrt{1x-\sqrt{2x- \cdots \sqrt{nx}}}$$ Then I looked at sequences of $f(x,1)^2,f(x,2)^2,\ldots,f(x,n_x)^2,f(x,n_x+1)^2,\ldots$ to observe, that for any small x there will be a $n_x$ from where the evaluations are no more complex but only real. Here are tables for the three initial values $x_1=0.1, x_2=0.01,x_3=0.001$ . It is interesting, that it seems, that the "critical" $n_x$ converges to some scalar multiple of the reciprocal of $x$ with decimal expansion of 216... . Hmmm....
For x_1=0.1
n f(0.1,n)^2
... ...
11 -0.302448089681-0.698792219012*I
12 -0.403301213973+0.649132465935*I
13 -0.262532045796-0.730589943250*I
14 -0.470664193786+0.569986543411*I
15 -0.116352885709-0.685312310620*I
16 -0.480413735006+0.369973112245*I
17 0.0771358518666-0.590686848231*I
18 -0.388211746572+0.195833371751*I
19 0.100000000000-0.291614520825*I
20 -0.181148810826
21 0.100000000000-0.131286629014*I
22 -0.0579158347766
23 0.0390823334499
24 -0.0157619545048
25 0.00542274237370
26 -0.00443384940800
For x_2=0.01
n f(0.01,n)^2
... ...
205 -0.114449346430-0.449020909239*I
206 -0.273308708016+0.245486912151*I
207 0.0100000000000-0.340544340711*I
208 -0.208411838263+0.166444378377*I
209 0.0100000000000-0.199665697307*I
210 -0.160212530941+0.0947222555124*I
211 0.0100000000000-0.115794748628*I
212 -0.102423208202
213 0.0100000000000-0.0608690577550*I
214 -0.0369471693906
215 0.0100000000000-0.0288680515620*I
216 -0.0136757225356
217 0.0100000000000-0.0107416751802*I
218 -0.00424432651020
219 0.00282037509906
220 -0.00108467768822
221 0.000551999344977
222 -0.000248398685385
223 0.000118131086199
224 -0.0000545149536090
225 0.0000254248494297
226 -0.0000117292180991
227 0.00000543517323693
228 -0.00000248279244384
For x_3=0.001 // internal computation precision: 1200 dec digits
n f(0.001,n)^2
... ...
2149 0.00100000000000-0.0771283223568*I
2150 -0.0593075793195+0.0404599076046*I
2151 0.00100000000000-0.0465223021845*I
2152 -0.0465304437990+0.0160979218450*I
2153 0.00100000000000-0.0254836686768*I
2154 -0.0190199767319
2155 0.00100000000000-0.0127892711274*I
2156 -0.00817903802372
2157 0.00100000000000-0.00611680141511*I
2158 -0.00341928431634
2159 0.00100000000000-0.00278386814867*I
2160 -0.00127422891686
2161 0.00100000000000-0.000989592316296*I
2162 -0.000393138741302
2163 0.000256090838303
2164 -0.000100925375352
2165 0.0000516413154610
2166 -0.0000235893686835
2167 0.0000113842433808
2168 -0.00000535253161006
2169 0.00000254701184424
2170 -0.00000120460035318
2171 0.000000571115136334
2172 -0.000000270334146946
Here is a picture of the trajectory for $x_3=0.001$ and increasing n. Using to little internal precision (200 dec digits in Pari/GP) made it appear, that this converged to the 3rd complex unitroot v (where $v^3=1$) but using 800 digits precision let it converge to something near zero. I've not yet analyzed this in more detail... (Update/Correction: in the picture I've incorrectly writen $f(x,n)$ instead of $f(x,n)^2$. I'll correct the image later)
For x_4=0.0001 // internal computation precision: 3600 dec digits
n f(0.0001,n)^2
... ...
21579 0.000100000000000-0.00210357411418*I
21580 -0.00136741030855
21581 0.000100000000000-0.00100044565776*I
21582 -0.000601677005986
21583 0.000100000000000-0.000467850547807*I
21584 -0.000244806521553
21585 0.000100000000000-0.000204301322925*I
21586 -0.0000860033335819
21587 0.000100000000000-0.0000411110536353*I
21588 -0.0000247254403485
21589 0.0000142141209815
21590 -0.00000608992862360
21591 0.00000302817937704
21592 -0.00000140739716572
21593 0.000000675808679202
21594 -0.000000319550497880
21595 0.000000152205000659
21596 -0.0000000722426625118
21597 0.0000000343450467856
21598 -0.0000000163147256218
21599 0.00000000775254705649
21600 -0.00000000368314354146
21601 0.00000000174991223509
21602 -0.000000000831348437546