Professor Kate Poirier | OL33 | Spring 2021

Category: Examples

Example: series representation of a function

Here are some notes I made last semester for WeBWorK: Series — Taylor and Maclaurin series — #10

This is a straightforward example, but it is somewhat long and requires the use of certain keys that make this approach work.

Remember that there is a second approach to finding the series representation of a function that does not require these keys: given a function and a center, you can always start taking derivatives to find the Taylor series. If you need to answer questions about that Taylor series, you will need to find a pattern for the Taylor coefficients, which is easy in some cases and not so easy in others.

Examples: infinite series and intuition

Convergent versus divergent series can sometimes be non-intuitive. 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.

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.

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