Talk:Taylor Series

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(Greg 5/23/13: new section)
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You can also point out that things like the small angle approximation <math> \sin\theta \approx \theta </math> 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.
You can also point out that things like the small angle approximation <math> \sin\theta \approx \theta </math> 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.
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== Greg 5/23/13 ==
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Moved around a lot of stuff. Created helper page on [[Convergence]] which would need to be revised/added onto. I moved the Taylor series for sine, cosine, and e^x into the MME after the derivation of the general form and of log(1+x). Because of this I need to rework (or also move?) the cos30 example in the Basic Description. I think an example like this is good for the Basic Description because it is the simplest and most general "use" of a Taylor series, but if I move the cosine Taylor series from Basic Description, I can't use it as an example. This is also problematic because there are a few images that go along with the example. Also - is the introduction too long? I have been slowly adding to it to make it more specific, but some of it probably could be moved into Basic Description as well. I added a section for error bound in the MME, but right now it is at the bottom of my priorities. I will figure out what parts of Why It's Interesting to hide (since the pi and e stuff is pretty technical and long). Also will add some applications - some examples of which Steve has sent me. Another simple physics application worth mentioning, I think, would be the small angle approximation, so I will think about that. [[User:Gbrown2|Gbrown2]] 14:27, 23 May 2013 (EDT)

Revision as of 14:27, 23 May 2013

Contents

Gene 7/15

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!


Abram 12/14

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.

Basic description

  • 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

Approximating e

  • 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.

Anna 11/5

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.

Anna 7/9

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  \sin\theta \approx \theta 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.

Greg 5/23/13

Moved around a lot of stuff. Created helper page on Convergence which would need to be revised/added onto. I moved the Taylor series for sine, cosine, and e^x into the MME after the derivation of the general form and of log(1+x). Because of this I need to rework (or also move?) the cos30 example in the Basic Description. I think an example like this is good for the Basic Description because it is the simplest and most general "use" of a Taylor series, but if I move the cosine Taylor series from Basic Description, I can't use it as an example. This is also problematic because there are a few images that go along with the example. Also - is the introduction too long? I have been slowly adding to it to make it more specific, but some of it probably could be moved into Basic Description as well. I added a section for error bound in the MME, but right now it is at the bottom of my priorities. I will figure out what parts of Why It's Interesting to hide (since the pi and e stuff is pretty technical and long). Also will add some applications - some examples of which Steve has sent me. Another simple physics application worth mentioning, I think, would be the small angle approximation, so I will think about that. Gbrown2 14:27, 23 May 2013 (EDT)

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