June 22 Information

Let ${\displaystyle a_{0}=0}$ and ${\displaystyle a_{n+1}={\sqrt {A+Ba_{n}}}.}$ Then, assuming convergence, we have ${\displaystyle a_{\infty }=\ell ={\frac {B+{\sqrt {B^{2}+4A}}}{2}}.}$ Thus, for ${\displaystyle A=B=1}$ we have ${\displaystyle \ell =\phi ,}$ for instance. Now, for ${\displaystyle A=B={\frac {1}{2}}}$ we have ${\displaystyle \ell =1,}$ and ${\displaystyle \lim _{n\to \infty }{\color {red}2}^{n}{\sqrt {\frac {\ell -a_{n}}{2}}}={\frac {\pi }{4}}.}$ My question would be with what constant to replace ${\displaystyle {\color {red}2}}$ in general, for different values of A and B, so that the limit in question should converge to a finite non-zero quantity. In other words, if ${\displaystyle f\left({\frac {1}{2}},{\frac {1}{2}}\right)=2,}$ what is the general formula for ${\displaystyle f(A,B)}$ ? Thank you. — 79.118.171.25 ( talk) 22:57, 22 June 2015 (UTC) reply

Apparently, ${\displaystyle f(1,1)={\sqrt {1+{\sqrt {5}}}}~,}$ and the limit in question is the square root of the Paris constant. — 79.118.171.25 ( talk) 03:07, 23 June 2015 (UTC) reply

I get ${\displaystyle f(A,B)={\sqrt {2\ell /B}}}$. My idea is to let ${\displaystyle x_{n}=\ell -a_{n}}$ and write its recurrence formula. The behavior for small ${\displaystyle x}$ is dictated by the linear term of its Maclaurin series. ${\displaystyle \ell -{\sqrt {A+B(\ell -x)}}=0+{\frac {B}{2\ell }}x+...}$ From there it's not hard to understand the behavior of ${\displaystyle {\sqrt {\frac {x_{n}}{2}}}}$ and find ${\displaystyle f(A,B)}$. Egnau ( talk) 03:40, 23 June 2015 (UTC) reply

I arrived just these past few minutes at the same conclusion, and wanted to post it, but was unable to connect. :-) Thanks ! — 79.113.226.120 ( talk) 03:53, 23 June 2015 (UTC) reply

And in general, for ${\displaystyle a_{n+1}={\sqrt[{m}]{A+Ba_{n}}}}$ we have ${\displaystyle f_{m}(A,B)={\sqrt {\frac {m~\lambda ^{m-1}}{B}}}~,}$ where ${\displaystyle \lambda }$ is a root of ${\displaystyle t^{m}=A+Bt.}$ — 79.113.226.120 ( talk) 10:08, 23 June 2015 (UTC) reply

Resolved