Comments on: A look at a few Tripos questions I

By: Analysis I: Lecture 14 « Theorem of the week

Analysis I: Lecture 14 « Theorem of the week — Mon, 18 Feb 2013 12:31:54 +0000

[...] Gowers wrote some blog posts about past Tripos questions. At the end of one of those posts, he wrote about what is for us Lemma [...]

By: PL

PL — Wed, 05 Dec 2012 04:41:29 +0000

Firstly, I’d like to apologize for not knowing LaTeX.

In the example of finding the radius of convergence, in part i, because the exponent of n is -2, wouldn’t the ratio test of the (n+1)st term over the nth term result in z(n^2)/[(n+1)^2] rather than x[(n+1)^2]/(n^2)? I’m not sure if I’m missing something or if this was just a typo. Thanks for taking the time to read this:)

By: Antony

Antony — Fri, 13 Jul 2012 15:01:50 +0000

I would like to add my voice to those who have said this series of questions has been invaluable. I am an OU student, so I don’t directly benefit from discussing Cambridge exam preparation, but there is an awful lot to learn and try to follow in the Mathematics and the arguments you lay out. I am sure your time is in short supply, but if you ever do a series like this again I will happily read along and rest assured will be very, very grateful!

By: torridtornado

torridtornado — Thu, 03 May 2012 18:57:54 +0000

Dear Mr Gowers,I am sure your discussion reaches out to a much wider audience than your own countrymen.We will be obliged if you, as a professional mathematician can post something like this at intervals.Thank you!

By: A look at a few Tripos questions IV « Gowers's Weblog

A look at a few Tripos questions IV « Gowers's Weblog — Wed, 02 May 2012 14:36:05 +0000

[...] post belongs to a series that began here. Next up is a question about [...]

By: How to tackle an exam question… | Degree of Freedom

How to tackle an exam question… | Degree of Freedom — Tue, 01 May 2012 09:15:12 +0000

[...] my attention to a sequence of posts on Tim Gowers’s excellent blog, in which Prof. Gowers lets us into his thought processes as he tackles Cambridge maths exam questions. Well worth a read, if only as an antidote to the habit of reading every exam question as a request [...]

By: Stones Cry Out - If they keep silent… » Things Heard: e219v1

Stones Cry Out - If they keep silent… » Things Heard: e219v1 — Mon, 30 Apr 2012 15:12:13 +0000

[...] blog series is a lot of fun. Here’s part 1, part2, and part [...]

By: Monday Highlights | Pseudo-Polymath

Monday Highlights | Pseudo-Polymath — Mon, 30 Apr 2012 15:11:46 +0000

[...] blog series is a lot of fun. Here’s part 1, part2, and part [...]

By: gowers

gowers — Sun, 29 Apr 2012 22:39:29 +0000

I don’t quite understand what you are saying, but here is a proof that the function when and 0 otherwise is not differentiable at 0. Consider the quantity . When it is 0, but when it is , which tends to infinity. So your statement that both limits are finite is simply false.

A small additional remark is that the limits from above and below have to be equal.

By: Piero Giacomelli (@pierogiacomelli)

Piero Giacomelli (@pierogiacomelli) — Sun, 29 Apr 2012 21:33:53 +0000

You say, “Considering you previous definition the following function is differentiable in zero.” However, that is not true. (Hint: can be negative.).

In this case you mean that if h \in \mathbb{R}, then f is diffrentialible in a point x if the limit for h->0^+ or h->0^- is finite, but again in the function above both limits are still finite and again the function is not continuos .

Sorry I meant it was always one of my concern about limit’s definition and for me the exception still remains.
Do not ment to be irritant or boring

By: espressonator

espressonator — Sun, 29 Apr 2012 18:35:48 +0000

Posted in my blog are my comments about Professor Gowers’ Model answers to the question (paraphrased here and in my post) “Prove that differentiability implies continuity.” To access my comments go to the following new blog: http://espressonator.wordpress.com/

By: gowers

gowers — Sun, 29 Apr 2012 09:53:29 +0000

You say, “Considering you previous definition the following function is differentiable in zero.” However, that is not true. (Hint: can be negative.)

By: Piero Giacomelli (@pierogiacomelli)

Piero Giacomelli (@pierogiacomelli) — Sat, 28 Apr 2012 21:24:52 +0000

Prove that if a function is differentiable at , then it is continuous there. This one is something that I have always struggle with. Considering you previous definition the following function is differentiable in zero but it is obviously not continuos in it

f(x) = 1 for all x >= 0
f(x) = 0 otherwise

By: A look at a few Tripos questions II « Gowers's Weblog

A look at a few Tripos questions II « Gowers's Weblog — Sat, 28 Apr 2012 10:11:18 +0000

[...] Gowers's Weblog Mathematics related discussions « A look at a few Tripos questions I [...]

By: gowers

gowers — Sat, 28 Apr 2012 08:19:05 +0000

Some weird things were going on with me too. I thought I had carelessly typed when I meant and it turned out that I had typed the right thing and it was coming out wrong — but not nonsensically wrong. It seems to be a very strange WordPress bug. I’ve found a fix for it, which is to insert a few spaces where I would ordinarily not need to.

By: typo service

typo service — Sat, 28 Apr 2012 08:09:25 +0000

Bizarrely, the “alt text” for the images of the rendered TeX read correctly, but the images themselves do not. So perhaps this is an issue with my own browser’s cache.

By: typo service

typo service — Sat, 28 Apr 2012 08:07:06 +0000

There are two "|z| = 1"s where I believe a "|z| < R" is intended: one in the paragraph (i) beginning the discussion of the radius of convergence, and significantly later in the paragraph beginning "Suppose, then, that you are explicitly asked to prove..."