e^pi > pi^e

Let's prove that $latex e^{\pi} > \pi^e$ without a calculator. If you haven't seen this before, give it a try before you read any further. Consider the function $latex f(x) = x/\ln(x)$ on the interval $latex (1,\infty)$. This function shoots off to positive infinity as $latex x$ tends towards either endpoint of the domain. Let's … Continue reading e^pi > pi^e

Limits and Towers

Consider the problem of finding the following limit:$latex \lim_{x \rightarrow 0} x^x&s=2$It's actually not too bad. We can write$latex x^x = e^{ x \ln x}&s=2$and bring the limit into the exponent (as exponentiating is continuous) to get that$latex \lim_{x \rightarrow 0} x^x = e^{\lim_{x \rightarrow 0} x \ln x}&s=2$From here, all we need to do … Continue reading Limits and Towers

Borwein Integrals

Feel up to seeing a bit of mathematical magic? See here.

Don’t Mess with the Mathematician

At the end of my undergraduate degree, I remember thinking "what better way to mark such an event than to write and record a maths rap?" Well, I did end up recording such a thing, but it was pretty woeful. However, a couple of months ago, I plucked up the zest to do it again. … Continue reading Don’t Mess with the Mathematician

Late night integration

Integrals can tell us quite a lot. For those of you who are so disgustingly bored that you've found your way onto my blog, you should have a go at evaluating exactly the following integral:$latex \int_0^1 \frac{x^4 (1-x)^4}{1+x^2} dx$It might take some effort, but it's well worth it. Of course, Wolfram Mathematica could do it … Continue reading Late night integration

Advice to a Young Mathematician

I thought it would be a good idea to relay some of the advice given by Sir Michael Francis Atiyah during his talk on Tuesday. Sir Michael Atiyah during his talk at #hlf13 Picture: HLFF @Bernhard Kreutzer Always ask yourself questions. Atiyah says that one of the secrets of his success is to always be curious. … Continue reading Advice to a Young Mathematician