Sadly, at least at my university, Taylor series is the topic that is least retained by students years after taking Calculus II. They can remember the rules for integration and differentiation, but their command of Taylor series seems to slip through the cracks. In my opinion, the reason for this lack of retention is completely understandable from a student’s perspective: Taylor series is usually the last topic covered in a semester, and so students learn them quickly for the final and quickly forget about them as soon as the final is over.
Of course, when I need to use Taylor series in an advanced course but my students have completely forgotten this prerequisite knowledge, I have to get them up to speed as soon as possible. Here’s the sequence that I use to accomplish this task. Covering this sequence usually takes me about 30 minutes of class time.
I should emphasize that I present this sequence in an inquiry-based format: I ask leading questions of my students so that the answers of my students are driving the lecture. In other words, I don’t ask my students to simply take dictation. It’s a little hard to describe a question-and-answer format in a blog, but I’ll attempt to do this below.
In the previous posts, I described how I lead students to the definition of the Maclaurin series
which converges to 
Step 7. Let’s now turn to trigonometric functions, starting with 
What’s 
As before, we continue until we find a pattern. Next, 
Next, 
Next, 
No pattern yet. Let’s keep going.
Next, 
Next, 
Next, 
Next, 
OK, it looks like we have a pattern… albeit more awkward than the patterns for 
If we stare at the pattern of terms long enough, we can write this more succinctly as
The 
Let’s see… 
In Precalculus, a function 
A pedagogical note: In my opinion, it’s better (for review purposes) to avoid the 
Step 8. Let’s now turn try 
What’s 
Next, 
Next, 
It looks like the same pattern of numbers as above, except shifted by one derivative. Let’s keep going.
Next, 
Next, 
Next, 
Next, 
OK, it looks like we have a pattern somewhat similar to that of $\sin x$, except only involving the even terms. I guess that shouldn’t be surprising since, from precalculus we know that 
Plugging into the series, we find that
If we stare at the pattern of terms long enough, we can write this more succinctly as
As we saw with 
For example, the series for 
using the periodicity of
Finally, using the cofunction identity 
In this way, the sine or cosine of any angle can be reduced to the sine or cosine of some angle between 
Step 10. For the final part of this review, let’s take a second look at the Taylor series
Just to be silly — for no apparent reason whatsoever, let’s replace 
![e^{ix} = \displaystyle 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \dots + i \left[\displaystyle x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \dots \right]](https://s0.wp.com/latex.php?latex=e%5E%7Bix%7D+%3D+%5Cdisplaystyle+1+-+%5Cfrac%7Bx%5E2%7D%7B2%21%7D+%2B+%5Cfrac%7Bx%5E4%7D%7B4%21%7D+-+%5Cfrac%7Bx%5E6%7D%7B6%21%7D+%5Cdots+%2B+i+%5Cleft%5B%5Cdisplaystyle+x+-+%5Cfrac%7Bx%5E3%7D%7B3%21%7D+%2B+%5Cfrac%7Bx%5E5%7D%7B5%21%7D+-+%5Cfrac%7Bx%5E7%7D%7B7%21%7D+%5Cdots+%5Cright%5D&bg=f3f3f3&fg=888888&s=0 "e^{ix} = \displaystyle 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \dots + i \left[\displaystyle x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \dots \right]")
after separating the terms that do and don’t have an 
Hmmmm… looks familiar….
So it makes sense to define
which is called Euler’s formula, thus proving an unexpected connected between
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