Originally posted by @blood-on-the-tracks
Fair enough.
Between you and me, I accept that my original explanation was hastily typed and clumsy. I don't so much accept the criticism of my reply to you, but if that is your perception, then I fair enough.
I do apologise for my tetchy reply to you.
All I would say, is that we would all like the Maths explained at a !evel that makes ...[text shortened]... r more aggressive ways.
I am assuming that you accept my rather long winded proof is correct?
Yes, it is correct. It was that factor of 1/2 that bothered me, once it dawned on me that it had been introduced into the numerator in the first step then the correctness of the proof became clear to me.
I did wonder if some sort of proof by iteration was possible. Since the n = 0 term is just 4/3. The problem is I can't see a way of proving the iteration in a way that doesn't exploit some working similar to either Duchess' or your method, in which case the iteration is redundant.
Bear in mind that people have to read this stuff off a screen, and without most of the formatting that happens in maths books, or even places like Wikipedia. Writing an integral in these forums is a monstrous pain. Quite often they'd understand the step if it was written down on paper, but the medium sort of gets in the way.
I can only think of two universities in this country that have entrance exams and I think one of them stopped using them. You went to Oxford?