Quote from: Brother Daniel on August 25, 2016, 11:38:32 AMSo does anyone want to see a proof of the irrationality of pi?(The one proof that I've read requires some basic differential calculus in order to follow it.)hit us up brah.
So does anyone want to see a proof of the irrationality of pi?(The one proof that I've read requires some basic differential calculus in order to follow it.)
Sorry, saw a shiny thing.
So yay for Ivan Niven. There's no way I'd ever be clever enough to think of such a proof myself.If anyone is unhappy with any of those steps, let me know. I may have hurried in a few places.
Mr. Knox. Now come now. Come now.You don't have to be so dumb now....Try to say this, Mr. Knox, please....Now if we look at the derivatives of g, we can see that g''(x) = b N Σ (-1) jπ 2N - 2jf (2j + 2)(x) where the sum is over j from 0 to N.We can shift the summation variable in order to write this as g''(x) = b N Σ (-1) j - 1π 2N - 2j + 2f (2j)(x) where the sum is now over j from 1 to N + 1.But we can throw away the j = N + 1 term from the sum, because we know that f (2N + 2)(x) = 0. So the sum really just takes j from 1 to N.And with a tiny rearrangement, we can write this as g''(x) = -π 2b N Σ (-1) jπ 2N - 2jf (2j)(x) where again the sum is over j from 1 to N.Comparing this expression for g'' to the expression for g, we can see that most of the terms cancel in the expression g''(x) + π 2g(x). This latter quantity is really just equal to π 2 times the first (j = 0) term of g(x). That is:g''(x) + π 2g(x) = π 2N + 2b Nf(x) = π 2a Nf(x)
(@ Vox)Uh oh.We may have a display problem.I put "p" in font tags, for the "symbol" font. I thought it would show up as pi for everyone.I don't know what you did to make a pi character just now. But whatever you did, maybe I should have done that instead.
Was it ok before that point?
Now f (k)(0) and f (k)(1) are both integers, for any nonnegative integer k.
Quote from: MikeS on August 25, 2016, 01:14:04 PMfypWas it ok before that point?
Yes, quite. One of my favorite books as a kid.
OH, you mean the proof. Hell if I know.