Proof that I don't get ItKlaproth says I ate your writeup Shit happens. Dated! Node Heaven will become its new residence.
Either my TV is broken and A. Schwarzenegger is already the President of the United States, or intense historical research has uncovered the deep humanity of the 1990s. Or I don't get this place.
(idea) by ariels (2 s) CC [ ] (print) (marked for destruction) Rep: 2 ( +12 / -10 ) Mon May 15 2000 at 14:15:40
Prevailing philosophy of the 1990s, regarding the importance of caring for others, and Man's responsibility towards his fellow Man. Was probably best expressed by the influential moral philosopher (who later went on to become the President of the United States) A. Schwarzenegger in some of his best works, produced in that period.
One of my interests is in physical proof, or even physical demonstration, of mathematical theorems. In many (but not all) cases these proofs can be made rigourous. But even when they cannot, they almost always give an interesting outlook on the theorem, helping understand its relevance or its veracity.
So far I've written up these; please do /msg me if you write or read any others.
- Stevens' physical proof of the sine rule
- (My) physical proof that the arithmetic mean is at least as large as the harmonic mean
- Demonstration of curvature
ariels' homenode gets top billing by E2's harshest critics!!1!
<Halspal> Now something needs to be done about excessively long homenodes. I'm getting on in years and it becomes difficult for me to grab that little elevator deal for super long homenodes.
Another puzzle: For n > 1 let f(n) be the number of distinct factorisations of n. Eg 12 has the factorisations 12, 2*6, 3*4, 2*2*3, so f(12) = 4. (And take f(1) = 1). Show that the sum of f(n)*n^(-2) from n = 1 to infinity is 2.
P = (1+1/22+1/24+1/26+...) * (1+1/32+1/34+1/36+...) * (1+1/42+1/44+1/46+...) * ...For instance, the contribution of n=12 (which should be 4/144) comes from picking the terms 1/122 (and "1" from all other series), 1/22*62, 1/32*1/42, and 1/24*32.
The only problem is to compute this product (and to prove that it converges).
By computing each geometric series, we have that
P = Πd≥2 d2/(d2-1)So
1/P = Πd≥2(d2-1)/d2 = Πd≥2(1-1/d2) = Πd≥2(1+1/d)(1-1/d) =But
Πd=2n (1+1/d) = Πd(d+1)/d = (n+1)/2and
Πd=2n (1-1/d) = Πd(d-1)/d = 1/nSo 1/P, which is the limit of the product of these two values, must be 1/2. Accordingly, P=2, as required.