## Continuity

I finally finished Analysis Block A today, and oddly enough I’m a bit sad to see it go. Unit AA4: Continuity was a lot more fun that I expected, and I particularly enjoyed the section about the Intermediate Value Theorem. The bit in the DVD programme about antipodal points blew my mind – I was convinced that it was impossible to prove that you can always find two antipodal points on the Earth’s equator that have the same temperature, right up until they explained why it must be true. I even made Alex come over and watch that section, it impressed me that much! (The similar examples involving balancing a table on bumpy ground, and equally distributing sugar on pancakes, were lots of fun too.)

TMA04 is wrapped up now, and I’m proud to say that this will be my first ever LaTex-produced assignment! Well, I should say Lyx-produced, because no matter how much I tried to get to grips with LaTex editors like TeXnicCenter, I just couldn’t get the hang of it, so I ended up resorting to a more WYSIWYG-style program instead.

I really enjoyed using Lyx to type up this assignment, so I reckon I’ll be sticking with this approach in future (though I wouldn’t rule out going back to TeXnicCenter or a similar editor – perhaps when I’ve gotten thoroughly used to Lyx itself). And I’d definitely recommend Lyx for anyone who’s interested in moving away from word-processing assignments; I think Lyx has just enough WYSIWYG elements to make it a relatively easy program to learn to use.

So now, at last, I can finally get on with Group Theory Block B – it’s been a long time, so let’s hope I haven’t forgotten everything from Block A!

## Sequences & Series

I found myself a bit underwhelmed by Unit AA2, to be honest. It was interesting, and I really enjoyed the part of the associated DVD program about approximating pi by taking exterior and interior polygons around a circle, but overall the unit didn’t quite grab me in the same way as the chapter in MS221 about sequences.

Unit AA3 was much more satisfying, and in a way it felt like AA2 was really just an introduction to the ideas we needed for AA3. For me, working with limits feels a lot more rewarding when it’s part of an attempt to prove that an infinite series converges/diverges. And the behaviour of infinite series is quite an interesting topic in itself – especially counter-intuitive results like the fact that $\sum_{n=0}^{\infty } 1/n^2$ is convergent but $\sum_{n=0}^{\infty } 1/n$ is divergent. I really do love results that make me go “but.. but how??” like that.

Still, even though AA3 was more enjoyable than AA2, I think the Analysis Block A has been a bit of a let-down compared to the first half of the course. I’m quite looking forward to unit AA4, as it’s about continuity, and I’m quite eager to find out how on earth continuity can be formally defined (since the only definition I’ve ever come across is the informal “you can draw the graph without picking your pen up” one), but my main aim at the moment is to make it through Analysis Block A so that I can enjoy the sweet, sweet goodness of Group Theory Block B. There are only three units of it, but I’m going to savour every page!