Still looking for any insightful ideas!
Hello.
Does anybody happen to know a closed form of this infinitely nested radical?
By any chance, maybe you even saw it somewhere?
I haven't had too much success so far. At the moment I am so desperate that I'm even willing to try and guess the solution, then prove that it is equal to my nested radical. For any real positive x the limit indeed exists (various criteria can be found for that very reason). Numerical limits can be seen in the plot:
Also here is a convergence plot:
It is made in a sense that using double precision variables computer sees no difference between a_{k}(x) and a_{k+1}(x) which in turn means that ~16 decimal digits have already been found. In fact it's so nasty that a{6}(50000) - a{5}(50000) < 10^(-24).
Notably the bigger my argument, the faster it converges (although I'm not sure what useful conclusions I can draw from that).
Pretty much the only known elegant cases: a(1) is equal to golden ratio, a(4)=2.
What would you suggest?
Best regards,
Pranas.
I'm not sure your equations are correct, but I think you mean that , and you would like to find .
If you can prove that the limit exists, then it's easy to determine what that limit is. Take the limit of the recurrence relation , giving . Then you can solve for a(x).
How do you prove that the limit exists? You can probably prove that , so it's bounded below. Then you would need to prove that it is decreasing: . And there's a theorem that a decreasing sequence that is bounded below always converges.
You might have to vary your approach for different ranges of x and make some assumptions about .
- Hollywood