From Math Images
I'd change your beginning sentence
- A Taylor series, or Taylor polynomial, is a function's polynomial expansion that approximates the value of this function around a certain point.
You should probably start out saying what a Taylor series is, since a Taylor polynomial is a different beast.
In the first equation you write "sin(x) =...", but for any n, it's just an approximation, so you want the "approximately equal" sign, not equality.
I made a couple tiny changes to the Basic description first paragraph.
Note that from your expression for log(x) you can calculate log(0) !
A small change in the first sentence of How to derive Taylor Series from a given function.
Please note that you can easily define what "infinitely differentiable" is, but "infinitely large" is a no-no. Out with it!
There's a lot of good content in this page. I've also got lots of comments/questions, one of which I think addresses your questions.
- This section doesn't say what a Taylor series is, just a Taylor polynomial, but the page is titled "Taylor series".
- Including the bit about periodic functions is fine, but only if you say a bit more, like "Taylor polynomials are normally used to approximate non-periodic functions. Periodic functions are more often approximated with Fourier series". Include a mouse-over def. of periodic and a red link to Fourier series.
- The sentence "Therefore, when..." is not a complete sentence.
- Referring to the animation is great, but only if it's explained what n represents. This would require generally expanding the intro, but I don't think that's so awful.
Example Taylor series
- I used the phrasing "The graph of this polynomial is shown in green in the image on the right, while the graph of the original function is shown in red" instead of the original phrasing because I felt like a Taylor series page is sufficiently advanced that we should be correct about using the term function versus graph of a function.
- I changed your phrasing about how the approximation becomes poor within 0.2 units of 1 because "poor" approximation seems so subjective, while saying that the difference can be seen in the graph is not.
- If these two new phrasings seem good to you, can you or I make similar changes in the other examples
- We've moved from using denominators expressed as factorials in the previous section to just using numbers in this section. Is that a problem?
Small angle approximation
- "Want to know where the equations come from..." is nice in its chattiness, except that it's so different in tone from the rest of the article.
- After "Let's calculate some values..." I think those equations need to be split into a few different lines. As written, it's a bit confusing.
- Can you define theta_max
- I changed the notation a little bit to just emphasize how we were using the Taylor series of the exponential function to approximate e. If that change seems ok, you or I can finish updating the notation.
This is my newest project. I have a couple questions to start out with--I feel like I want to specify that taylor polynomials should be used with non-periodic functions, but then I feel like I've got to created a page on Fourier series, which is what you use in the periodic case. Not that that would be hard or take long, I just wanted your opinion.
Can you maybe add a short paragraph on why we use them? Polynomials are super easy to deal with, which is why we love approximating everything with them, and even just stating that could be a big help.
You can also point out that things like the small angle approximation is really just a first order taylor series, and without that, we can't even solve the equation of a simple pendulum. Anyways... that would just be a really simple example to give a bit if context to the page.