On 07/10/2016 00:56, Dan Asimov wrote:
Then try this: Part II of last June's Mathematical Tripos (U.K.): [pure + applied] [math + physics]:
https://www.maths.cam.ac.uk/system/files/list_ii.pdf <https://www.maths.cam.ac.uk/system/files/list_ii.pdf>
—Dan
P.S. I got so tired reading it I had to lie down until the feeling passed.
P.P.S. At least I can solve problem 2H on page 99. Can you?
Yes. ... SPOILER SPACE ... ... SPOILER SPACE ... ... SPOILER SPACE ... ... SPOILER SPACE ... ... SPOILER SPACE ... ... SPOILER SPACE ... ... SPOILER SPACE ... ... SPOILER SPACE ... ... SPOILER SPACE ... ... SPOILER SPACE ... This is a simple generalization of a standard proof of the irrationality of e (the special case where all the a's equal 1). Suppose our sum S equals p/q and choose n = (M+2)q, say. Then n!S is an integer. Divide the terms in the sum into those with denominator <= n! and those with denominator > n!. The former again yield integers. The latter aren't all zero (because infinitely many a's are nonzero); their sum has the same sign as the first nonzero one (because the denominators grow so fast that |each term| > sum of |later terms|) and in particular is not zero; and its absolute value is < 1 (by comparison with a geometric series). But, alas, no integer strictly between 0 and 1 has ever been found and scientists fear none exists, in which case we have a contradiction. (Or, a little higher-brow: rational numbers cannot be very well approximated by other rational numbers, and the series converges fast enough to contradict this. But I think that's a little more work.) It may perhaps be worth mentioning that no one is expected to learn all the material in these courses. A good student might attempt a quarter of them, probably with some thematic consistency. An exceptional student might attempt most of the pure or most of the applied but would know some much better than others. -- g