4 ms·
Right. Also (just a few more concrete examples): • the sum of the series 4(1 - 1/3 + 1/5 - 1/7 + …) will still be our π: https://en.wikipedia.org/wiki/Leibniz_
by svat 3y ago
Right. Also (just a few more concrete examples):
• the sum of the series 4(1 - 1/3 + 1/5 - 1/7 + …) will still be our π: https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80 https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80
• the sum of the series (1 + 1/4 + 1/9 + 1/16 + 1/25 + …) will still be π²/6: https://en.wikipedia.org/wiki/Basel_problem https://en.wikipedia.org/wiki/Basel_problem
• (therefore) the probability that two numbers chosen uniformly at random from [1…N] are relatively prime will still approach 6/π² as N grows large
• the product 2(4/3)(16/15)(36/35)(64/63)(100/99)… will still be our π: https://en.wikipedia.org/wiki/Wallis_product https://en.wikipedia.org/wiki/Wallis_product
• the value of (n!/(√n (n/e)^n))²/2 as n grows large will still (very slowly) approach π: https://en.wikipedia.org/wiki/Stirling%27s_approximation https://en.wikipedia.org/wiki/Stirling%27s_approximation (e.g. https://www.wolframalpha.com/input?i2d=true&i=N%5C%2891%29Divide%5B%5C%2840%29Divide%5BN%5C%2891%29n%21%5C%2893%29%2C%5C%2840%29%E2%88%9An+Power%5B%5C%2840%29Divide%5Bn%2Ce%5D%5C%2841%29%2Cn%5D%5C%2841%29%5D%5C%2841%29%C2%B2%2C2%5D%5C%2893%29+for+n+%3D+10000 https://www.wolframalpha.com/input?i2d=true&i=N%5C%2891%29Di... )
and so on, for most of the non-geometry results listed: https://en.wikipedia.org/w/index.php?title=List_of_formulae_involving_%CF%80&oldid=1179793348#Formulae_yielding_%CF%80 https://en.wikipedia.org/w/index.php?title=List_of_formulae_...
- NeoTar 3y agoIn a change from the normal refrain of 'there's an XKCD about that' - in this case there is an Saturday Morning Breakfast Cereal (SMBC) about it: https://www.smbc-comics.com/comic/pi-2?ref=refind https://www.smbc-comics.com/comic/pi-2?ref=refind For those unwilling to click-through, it essentially posits an alternate history where infinite series were explored by mathematicians before geometry, so rather than being surprised that the 'circle constant' is found in many infinite series, we would instead be surprised that the 'infinite series constant' is found in the geometry of a circle.
- thechao 3y agoPi is the scaling factor of the diameter of a circle to its circumference; there's an infinite set of such scaling factors: one for each ellipse (the circle is a special case). I wonder which/what sort of infinite series arise from/for the generalized elliptic scaling factors?
- NeoTar 3y agoI rather suspect that the generalised elliptic scaling factor is a continuous function, so the answer may be a bit boring. For any infinite series with a finite sum I would be able to give you an ellipse (indeed, probably an infinite number of ellipses) whose scaling factor is a rational multiple of the sum.
- thomasahle 3y ago• The probability that a random walk with n steps return to the origin is 1 - π/logn + O(1/logn)^2: https://twitter.com/thomasahle/status/1719140649952571672 https://twitter.com/thomasahle/status/1719140649952571672