A look at a few Tripos questions II

This is the second in a series of posts that started here. In the first post I explained what I’m up to. Now let me just continue with some more questions. I’m now on to the harder Section II questions. Here’s the first one I want to look at. Even though it makes the posts shortish, I think I’m going to stick to one long question per post.

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9F. Prove the Axiom of Archimedes.

Let be a real number in and let be positive integers. Show that the limit

exists, and that its value depends on whether is rational or irrational.

[You may assume standard properties of the cosine function provided they are clearly stated.]

A look at a few Tripos questions I

When I was a mathematics undergraduate, I became aware of a huge cultural difference between mathematicians and engineers. That sounds like the beginning of a joke you’ve heard twenty times already, but it isn’t. The difference was that when mathematicians were set questions, they were expected to work out how to solve them, and if they couldn’t do so then it was too bad — the best they could do about it was ask their supervisors. But engineers had model answers for everything, available with the latest technology, which in those days was microfiche. In case you have no idea what I’m talking about, the answers were reduced in size by a factor of about five in each direction and printed on to some kind of transparent plastic that you could look at through a magnifying machine. There were a couple of the machines in our college library, and they were nearly always in use.

Model answers have always seemed to me to be a bad idea in mathematics, because it is hard to learn how to think for yourself when you are given the answers to all the problems you tackle. So it might seem a bit odd that in this post I’m going to attempt to help people preparing for Part IA of the Cambridge Mathematical Tripos by providing some model answers.