Creating digits of pi

[EdS2]

(02-17-2018 03:02 PM)Gerson W. Barbosa Wrote:  

(02-17-2018 12:27 PM)EdS2 Wrote:  I just came across this nice approximation, by Ramanujan (of course)
∜(2143/22) = 3.14159265258...

That's probably the only Ramanujan approximation that doesn't rely on any of his astounding theories. He just noticed that \(\pi ^{4}\) = 97.409091034..., very close to 97.409090909..., which when multiplied by 990 this gives 96435.

Hmm, would a mathematical genius look at a decimal expansion? I would hope for some rather more sophisticated source of the insight - but do we know, or can we ever know, where this approximation came from?

I can't resist sharing this other one from Ramanujan, which agrees to 18 digits apparently but with only 12 digits in the expression:

\(\pi \approx \frac{12}{\sqrt{190}}\log\big((2\sqrt{2}+\sqrt{10})(3+\sqrt{10})\big)\)

As continued fractions:
3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...
vs
3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 1, 2, ...

Edit: found this one in Ramanujan's papers - see TABLE II here.