Another Ramanujan Trick

[Valentin Albillo]

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Hi, Gerson:

(09-29-2019 11:19 AM)Gerson W. Barbosa Wrote:  You might want to check this out:\[\ln\left(\sqrt[4]{\frac{{305}^{3}+{25}^{3}+{5}^{3}+5+\frac{33}{{9}^{5}+{5}^{5}+{3}^{5}+3}}{99}} \right )\]Or that one:\[\sqrt[4]{\frac{2143+{\left(6+\sqrt{\frac{6}{1+{6}^{-6}}}\right)}^{-6}}{22}}\]

Nice, they're indeed aesthetically pleasing but as approximations to Pi they use far too many digits, functions and operations for the accuracy they achieve.

On the other hand, the following theoretically-based approximation to Pi is much better in that regard: just ten individual digits, two functions and two arithmetic operations to achieve almost 17 digits (save for 2 units in the last place):\[ \frac{3 Ln 640320}{\sqrt{163}} =3,1415926535897930...
\]
A small modification, essentially adding just three digits and one sum gets you almost 32 digits (again save for 2 units in the last place).\[ \frac{Ln{({ 640320}^{3}+744})}{\sqrt{163}} =3,1415926535897932384626433832797...
\]Regards.
V.
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