Pzi
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Hello.
Does anybody happen to know a closed form of this infinitely nested radical?
http://imageshack.us/a/img268/6544/radicals.jpg
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:
http://imageshack.us/a/img842/876/plote.jpg
Also here is a convergence plot:
http://img100.imageshack.us/img100/64/convergence.jpg
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?
Pranas.
Does anybody happen to know a closed form of this infinitely nested radical?
http://imageshack.us/a/img268/6544/radicals.jpg
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:
http://imageshack.us/a/img842/876/plote.jpg
Also here is a convergence plot:
http://img100.imageshack.us/img100/64/convergence.jpg
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?
Pranas.
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