On a particular continued fraction related to $\pi$.

Asked May 16 '16 at 09:08

Active May 16 '16 at 09:36

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Is this continued fraction well studied?

$$\Theta(m)=m+\cfrac{1^2}{2m+\cfrac{3^2} {2m+\cfrac{5^2}{2m +\cfrac{7^2}{2m+\cfrac{9^2}{2m+\ddots}}}}}$$

Note $\Theta(1)=\frac4\pi$.

Denote $\Theta(m)=\frac4{c_m}$.

With this notation $c_1=\pi$ which has an AGM procedure.

Is there an arithmetic geometric mean technique general $c_m$?

edited May 16 '16 at 09:36

asked May 16 '16 at 09:08

Turbo

1

Why are there two parameters to your function when the only way they're used in the definition is multiplied together? – hmakholm left over Monica May 16 '16 at 09:14
Why do you need an AGM procedure for this, the continued fraction seems to be a better way to compute this function – Yuriy S May 16 '16 at 10:37
AGM is faster much much faster that is why Gauss invented it. – Turbo May 16 '16 at 10:50
@Turbo, it involves taking square root. It can be much slower if you do it by hand – Yuriy S May 16 '16 at 10:53
@Yuriy that would be fine. AGM is fastest technique for pi provided we have unlimited word size. – Turbo May 16 '16 at 10:53
@Turbo, see this paper, I think it's your continued fraction exactly – Yuriy S May 16 '16 at 10:57
@YuriyS you mean one in abstract? no. I have seen this. It does not help. – Turbo May 16 '16 at 10:58
This CF is evaluated here. – metamorphy Jul 15 '20 at 01:38

0 Answers0