4 ms·
Yes! Not only that, here's something I found mind-blowing: for almost all real numbers, the nth root of the nth convergent's denominator has as limit the same v
by svat 4y ago
Yes! Not only that, here's something I found mind-blowing: for almost all real numbers, the nth root of the nth convergent's denominator has as limit the same value, and that value is e^(pi^2/12ln2).
Am typing from phone so can't write it down here properly, but some details at an old blog post of mine: https://shreevatsa.wordpress.com/2010/04/30/some-incredibly-beautiful-things/ https://shreevatsa.wordpress.com/2010/04/30/some-incredibly-...
- madcaptenor 4y agothere's also Khinchin's constant - for almost all real numbers, the geometric mean of the first n continued fraction coefficients approaches a constant K as ngoes to infinity. K is about 2.68 (the geometric mean of the Gauss-Kuzmin distribution). I like to pair this with a more trivial fact - for almost all real numbers, the arithmetic mean of the first n digits of the decimal expansion, as n goes to infinity, approaches 4.5.
- tialaramex 4y ago> for almost all real numbers, the arithmetic mean of the first n digits of the decimal expansion, as n goes to infinity, approaches 4.5. Is this just because Almost All Reals are Normal, and so in every base they have an equal distribution of all digits ?
- madcaptenor 4y agoThat’s exactly it.
- cperciva 4y agoHere's an even more trivial fact: For almost all real numbers, the geometric mean of the first N digits of the decimal expansion, as N goes to infinity, approaches zero. (Really really fast!)
- madcaptenor 4y agoand a less trivial one: the arithmetic mean of the continued fraction coefficients goes to infinity.