During this week’s COLD session we talked a little about the intuition behind convergent versus divergent series. The tricky part is, that a lot of this is very unintuitive, so we really have to rely on our tools (convergence tests) in order to understand what’s going on.

In one sense, the convergent series are the weird ones. If you add up infinitely many positive numbers, for example, why should you expect your sum to be anything other than infinity? Xeno’s paradox really confusing when you don’t have the language of limits. But we know that the geometric series

\[ \sum_{n=1}^\infty \left(\frac{1}{2} \right)^n\]

converges to 1. Remember that convergence is a verrrrrry special property. If you were to just bump into a random series on the street, it is very unlikely that it would converge.

In another sense, the divergent series are the weird ones, even though they’re way more common. Just like how there are different reasons that a limit can fail to exist, there are different ways that a series can diverge. Again, this can be very unintuitive, so if you can’t wrap your head around it right away, you’re not alone!

Here are some videos that might help you start wrapping your head around these ideas…but they might also just blow your mind. Don’t worry if that’s that’s the case. They don’t just show calculations, like most of the videos on the course hub do, but they get more into the concepts underlying those calculations.

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This first video discusses Xeno’s paradox again, in a way you might not have seen before.

This next video features the Fields Medalist, Charles Fefferman. There is no Nobel Prize for mathematics (and rumors around why, including one about Alfred Nobel’s wife having an affair with a mathematician). The Fields Medal one of the biggest prizes in math and plays the role of the Nobel in the field.

The next two videos discuss some wacky things that can happen when you try to determine the sum of a divergent series.

The next video features mathematician Edward Frenkel. He gets into how divergent series can be manipulated so that they can still provide you with information. Fun fact: my office was just down the hall from Professor Frenkel’s office for a few years!

By the way, if you like these videos, there are waaaaay more over on the Numberphile YouTube channel.

I finally figured out how to approach the Taylor series question Katie asked this morning! The good news is: as I said this morning, once we had the key, it would be easy to unlock the rest of the problem. The bad news is: “easy” is a relative term; this still ended up being long. (Tip: this is probably too long a question to give on a test.) The other good news is that even though there are many steps, none of them require a huge leap.

The link to the notes is here and the document is also in the Dropbox folder containing COLD session notes. Apologies, but it’s unlikely I’ll be able to record a video of this one this week (maybe ever), so the notes will have to do for now. Please let me know if you have any questions about it.

Hopefully the color coding makes sense. I was going to skip finding the interval of convergence, but it was actually not too bad to track how the I of C changes as we manipulate the functions, so I snuck it in afterward in the appropriate places in brown.

You’ll notice that I split this into two warmup exercises before getting to the actual question that was asked. If you just want to see how the warmup exercises were used to find the series, skip to the last page.

Good luck!