Another Ramanujan Trick

[Gerson W. Barbosa]

(09-29-2019 09:36 PM)Valentin Albillo Wrote:  

(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).\[ \frac{Ln{({ 640320}^{3}+744})}{\sqrt{163}} =3,1415926535897932384626433832797...
\]

Hello, Valentin,

My second approximation uses 10 digits and produces 17 significant digits, but it requires four functions, twice as much compared to the first approximation you mention. By no means I intend to compete with Ramanujan, Dedekind and others. That was just an idle Sunday morning play.
Anyway, thanks for bringing these really nice approximations to my attention again. They have prompted me to search for an 82-digit one belonging to the same group. I’ve finally found it in Tito Pieza’s blog here.

Best regards,

Gerson.