Comments on: A look at a few Tripos questions II

By: Greg Marks

Greg Marks — Mon, 30 Apr 2012 21:03:38 +0000

Although the statement “its value depends on whether $x$ is rational or irrational” is not quite the same as saying “its value depends only on whether $x$ is rational or irrational,” it is close enough that students might consider the possibility that plugging in the simplest rational $x$ they can think of reveals what they’re supposed to prove. (This sort of ploy was also helpful, for example, on Problem A2 of the 2011 Putnam Competition.)

By: gowers

gowers — Mon, 30 Apr 2012 14:04:22 +0000

Yes it is correct. If you want a simple proof, just let be an arbitrary sequence of 0s and 1s and take the number . If you multiply that by and take the integer part, you’ll get something close to . Therefore, it’s easy to make sure that doesn’t tend to a limit. If you want not to tend to a limit, just take the previous and divide it by 2.

By: Ether

Ether — Mon, 30 Apr 2012 11:17:37 +0000

I remember doing this question (as a practice) and wondering ‘What happens if you swap the limits around’.

As in, you let m tend to infinity and then n tend to infinity. I think that for a irrational number it can fail to have a limit. Is that correct?

By: Sean

Sean — Sat, 28 Apr 2012 22:43:08 +0000

A large part of solving an unseen question (and doing mathematics in general) is figuring out what method to use or what lemmata to prove without being told these in advance, and without necessarily having confirmation at every step of the way that one is on the right track.

I’m guessing the examiners want to test this skill, though it’s hard to do this if the questions are too prescriptive. Even a statement saying which assumptions are allowed would probably be interpreted as a significant hint that the student should make use of those assumptions.

By: David

David — Sat, 28 Apr 2012 21:25:35 +0000

This is what you get for not reading the whole post before commenting…

By: David

David — Sat, 28 Apr 2012 21:23:43 +0000

I think there’s a slight error in your characterisation of when cos^2(m!*pi*x) is 1: m!*pi*x could also be an odd multiple of pi.

By: pierre

pierre — Sat, 28 Apr 2012 19:18:42 +0000

It seems like things would be easier for everyone if the questions were written in a way that made clearer what could be assumed. For example, this question might be rewritten as:

1. Prove the Axiom of Archimedes.

2. Prove that t^n -> 0 whenever 0<=t<1 using the Axiom of Archimedes.

3. Using your result in 2, prove the following: let x be a real number in and let m,n …

It seems like a significant portion of this blogpost is dedicated to divining the mind of the examiners about allowed assumptions. Perhaps if the questions were written in a way that makes this clear everyone (that is, both you and the students) could spend more time focusing on mathematics.

By: gowers

gowers — Sat, 28 Apr 2012 16:07:08 +0000

The emphasis on “equals” there is slightly misleading: the really important thing is that all that matters is that the closeness happens when is sufficiently large — the fact that we get equality is a little bonus that we could in theory have done without.

By: gowers

gowers — Sat, 28 Apr 2012 16:05:41 +0000

I was being a bit brief there (as is appropriate if you’re in the middle of an exam). But the point is that we need to show that if is sufficiently large then the function is close to its limit. So if we prove that it actually equals the limit whenever then we’re done.

By: Nyaya

Nyaya — Sat, 28 Apr 2012 15:43:47 +0000

For the sake of completeness, shouldn’t one also mention the case when $x$ is rational and $m

On a different note, I envy your students for having such a great teacher.