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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 $A=B={\frac {1}{2}}$ we have $\ell =1,$ and $\lim _{n\to \infty }{\color {red}2}^{n}{\sqrt {\frac {\ell -a_{n}}{2}}}={\frac {\pi }{4}}.$ My question would be with what constant to replace ${\color {red}2}$ 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 $f\left({\frac {1}{2}},{\frac {1}{2}}\right)=2,$ what is the general formula for $f(A,B)$ ? Thank you. — 79.118.171.25 (talk) 22:57, 22 June 2015 (UTC)[reply]

Apparently,

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

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)[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