Wikipedia:Reference desk/Archives/Mathematics/2015 June 22

= June 22 =

Generalization of a Certain Formula for Pi
Let $$a_0=0$$ and $$a_{n+1}=\sqrt{A+Ba_n}.$$ Then, assuming convergence, we have $$a_\infty=\ell=\frac{B+\sqrt{B^2+4A}}2.$$ Thus, for $$A=B=1$$ we have $$\ell=\phi,$$ for instance. Now, for $$A=B=\frac12$$ we have $$\ell=1,$$ and $$\lim_{n\to\infty}{\color{red}2}^n\sqrt{\frac{\ell-a_n}2}=\frac\pi4.$$ My question would be with what constant to replace $${\color{red}2}$$ 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 $$f\left(\frac12,\frac12\right)=2,$$ what is the general formula for $$f(A,B)$$ ? Thank you. — 79.118.171.25 (talk) 22:57, 22 June 2015 (UTC)
 * Apparently, $$f(1,1)=\sqrt{1+\sqrt5}~,$$ and the limit in question is the square root of the Paris constant. — 79.118.171.25 (talk) 03:07, 23 June 2015 (UTC)
 * I get $$f(A,B) = \sqrt{2\ell/B}$$. My idea is to let $$x_n = \ell - a_n$$ and write its recurrence formula. The behavior for small $$x$$ is dictated by the linear term of its Maclaurin series. $$\ell - \sqrt{A + B(\ell - x)} = 0 + \frac{B}{2 \ell} x + ...$$ From there it's not hard to understand the behavior of $$\sqrt{\frac{x_n}{2}}$$ and find $$f(A,B)$$. Egnau (talk) 03:40, 23 June 2015 (UTC)
 * 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)
 * And in general, for $$a_{n+1}=\sqrt[m]{A+Ba_n}$$ we have $$f_m(A,B)=\sqrt{\frac{m~\lambda^{m-1}}B}~,$$ where $$\lambda$$ is a root of $$t^m=A+Bt.$$ — 79.113.226.120 (talk) 10:08, 23 June 2015 (UTC)