A fun introductory Taylor series exercise with Sin(x) I saw on Twitter last week

Last week I saw a neat “fun math fact” via a Matt Enlow tweet:

By happy coincidence my older son is studying Taylor Series this week, so I thought it would be fun to talk through the problem.

Here’s the introduction:

My son had some nice ideas about how to approach the problem in the last video, so next we went to the white board to work out the details:

Finally, I asked my son to finish up the details and then asked him for a sort of number theory proof of why 180 multiplied by an integer with all digits equal to 5 was always close to a power of 10:

Definitely a fun little problem – definitely accessible to students learning some introductory calculus.

An introductory talk about power series with my son and a surprise (to me) misconception

I’m wrapping up sequences and series this week with my son and the final topic is Taylor Series. We’d had a few discussions here and there about power series, but it all comes together this week. Looking through some old problems form the BC calculus test, I found a nice one from 2011 that I wanted to use to introduce the idea of error terms.

I intended for the first three parts to be review, but one interesting misconception came up – so the talk was more than just review.

Here’s the introduction to the problem and my son’s work on the first part of the problem. This problem asks you to write down the usual series for \sin(x) and then write down the series for \sin(x^2)

The next question asks you to write the series for \cos(x) and then write the Taylor Series around x = 0 for the function \cos(x) + \sin(x^2).

Here my son wrote the series for the 2nd function in a way that surprised me:

Once we wrote the correct series for the 2nd part of the last question, we moved on to part (c) of the problem -> find the 6th derivative of the function above evaluated at x = 0:

Finally, we looked at the last part of the problem. The question is about the error in a Taylor series approximation. I’d hoped to use this question to introduce ideas about error terms in Taylor Series, but unfortunately I completely butchered the discussion. Oh well – we’ll be covering the ideas here in a much more detailed way later this week: