From Math Images
Checklist for Writing Pages
Added by Greg Brown.
Messages to the Future
- This page is very large and it would be a stretch to add more to it.
- Taylor series as a topic can be expanded upon on Math Images, though. I talk about this more specifically in the Pendulum section of this discussion page. I ended up removing the pendulum content because it didn't directly pertain to Taylor series, but if content on composition/inversion of Taylor series were added, it would lead to some interesting options. This would need to be on some sort of "advanced Taylor series" page.
- Also Taylor series in several variables.
- The justification for the above is that certain pages, like Rope around the Earth and possibly Xintong's Systems of Linear Differential Equations, do refer to topics pertaining to Taylor series that are not featured on this page.
- Somewhat separately, Steve said he had "lots of" comments on the new error formula proof, but he never sent them to me. It is very likely that the statement of the proof could be improved, since I've done little revision of it. It would also be possible to reinstate the older error formula proof, which is correct technically speaking but is not as intuitive. That would, however, require some revision of the error formula section outside of the proof because the formula is stated slightly differently for the two proofs.
Reference and Footnotes
- All images are cited. Most generated either by me or Peng.
- Textual references are done.
- The page, I think, does a good job of this. It presents a few applications.
- The page doesn't exactly have a "hook." The main image is pretty compelling in my opinion, but by the end of the Basic Description, the only thing that constitutes a "hook" is the assertion that series can be used to approximate things we couldn't otherwise approximate.
- I imagine that not all readers will find this interesting, but it is a very mathy page, a kind of image/helper page hybrid, and there isn't a whole lot to say without going into the math.
Quality of Prose and Page Structure
- Opening paragraph is clear. Not quite as brief as most opening paragraphs on Math Images, but in the case of Taylor series, I think that the brief, sentence-long introduction of the main image should stay in the opening paragraph.
- The sequence of sections makes sense. The Basic Description tells what Taylor series are. The MME shows how to use them, how to find the general formula, how to apply the general formula, how to use Taylor series (mathematically), and how to bound error.
- Created Convergence helper page for a lot of old content that was too general for Taylor series. The new helper page clearly defines series, sequences, and what it means for them to converge.
- Error is placed last, since it's the most complicated and difficult topic on the page. It is also hidden, and the error formula proof within the error section is also hidden.
Integration of Images and Text
- Lots of appropriate, helpful images.
- Just about every image is referred to in-text, and its relevance is pointed out.
Connection to Other Mathematical Topics
- The page does a good job of handling the derivation of constants (e and π).
- Otherwise, it does make a lot of assertions about how Taylor series are used. I mention a couple times that physicists use Taylor series to approximate things that don't have closed-form solutions, but there isn't a really good example of this on the page. The problem with the pendulum example was that it had relatively little to do with Taylor series.
- Now, I mention the relevance of Taylor series for the pendulum solution and refer to the Simple Harmonic Motion page. I also refer to the Rope around the Earth page, which uses series approximations, although those approximations are slightly more complicated than what the Taylor Series page shows you how to do. I also refer to Xintong's page, when he uses Taylor series to calculate eA</math> (e to a matrix), but neither of us have actually justified that operation.
- I think this is a good compromise, since most of these topics just use Taylor series for a single step.
Examples, Calculations, Applications, Proofs
- Good amount of examples in several sections. I show how to do calculations and try to provide several different possibilities.
- I discuss applications above; there are not a lot of specific applications, but there are references to other pages that use Taylor series.
- Important general formulas are all proved with a good degree of rigor. There are also detailed derivations of more specific series.
Mathematical Accuracy and Precision of Language
- Language and notation are accurate. I think most of it is pretty accessible. The language may get hard to follow in the error section, but that is inevitable and due to the difficulty of the content rather than the presentation.
- Again, the proof of the error formula is the one part that may need more revision in this department. I am not sure if everything is stated right (although the math is correct). Particularly, I refer to the interval [a, x], a being the base point of the Taylor series and x being the value at which the power series is evaluated. But x is also the variable I am differentiating with respect to.
- Definitions are clear.
- Paragraph size, image spacing, etc. all seem fine.
- Relevant mouse-overs and links to other topics.
- The really difficult math is hidden.
- I see two small issues with images. When I do not full-screen the page, there are two arrays of 3 images that stretch the margins (in the sections The General Form of a Taylor Series and Small-Angle Approximation). The other issue is that there is one line of LaTeX math in the error proof formula that is also long enough to stretch the margins of some browsers. However that can pretty easily be fixed. With respect to the two arrays, I could try breaking them up?
My detailed review can be found here; the major comments from that document are included in the appropriate sections below.
Also, a new general comment since that document: You should capitalize all words other than articles and minor prepositions in your section titles.
-Diana 10:49, 15 July 2013 (EDT)
Steve M 6/10/13
- I'm doing a final "fine points" editing but it is taking time; there are a number of places where math words are not used right, or some side remarks are so vague they are best not included at all. I am about halfway done. I thought it would take less time to do them myself than to write them out legibly on paper for Greg, but I may be wrong.
- One important fine point: the verb form of the math noun "derivative" is not "derive" but rather "differentiate". This is because "derive" already has a different meaning in math - draw the logical conclusion.
- Also, problems have solutions, but expressions do not have solutions; they only have evaluations. Thus you can't solve an expression like . Instead, you evaluate it.
Steve M 5/31/13
- Some local editing that I will do right in the body of the document. Compare histories with Greg's version to see what I changed.
- 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)
Feedback by Section
Somewhere in the basic description, early on, you should say something (perhaps a simple verbal summary of the process) about where these series come from, and how they’re related to the functions they approximate. Readers who have never encountered this idea before may not understand that the series are generally constructed based on those functions, and are not arbitrary or universal.
Bring the end of the basic description, starting at "Readers may, without knowing it..." into the more mathematical explanation. The exploration of cosine may not even be necessary, but you could include it in the “Finding the Taylor Series for a Specific Function” section, before the ln example.
Since a lot of the basic description needs to go into the more mathematical section, you should expand on the idea that "Taylor series are important because they allow us to compute functions that cannot be computed directly" a bit more. Many students won’t understand why this would be useful for functions like sine – they commonly think of trig functions as numbers or some sort of weird multiplication operations. Bringing some of your later discussion of how calculators use similar approximations into this section would be good.
Can you say more about the paragraph, "We can compare this to the value.... for all displayed decimal places."? This is a great opportunity to make a connection for the reader and show the utility and ubiquity of these series. (And I suggest keeping that particular idea in the Basic Description.)
-Diana 10:49, 15 July 2013 (EDT)
- the Taylor expansion for sin(x) says "n varies from 0 to 36", which is consistent with the animation; however, the mathematical expression has (-1)(n-1)/2 and n actually should be odd. Maybe add a little bit of explanation there? (You mentioned this later in the page, but I think it's necessary here as well)
A More Mathematical Explanation
Basic use of Taylor series
The general form of a Taylor series
Finding the Taylor series for a specific function
- "By using the identity log(a ·b ) = log (a ) + log (b )": since the paragraph is mostly concerned about natural logs, it might be clearer to just write ln(a · b) = ln (a) + ln(b)
- "Other Taylor series, to be introduced in the next section, are absolutely convergent; they converge for all x.": mouse-over for "absolutely convergent" may be necessary
Other Taylor series
- "Euler's number" link -> "Euler's Number"
- "For instance, we can compute the Taylor series of the function composition ": use inline format for sin(2x) and 2x2
- "although a given Taylor polynomial for some finite n may not be accurate" huh? T Polys are rarely accurate if you measure close enough.
- For e^sin x "one may substitute the whole Taylor series of sin(x) for x in the Taylor series for e^x". Show me. Why is it useful?
- Last sentence of subsection needs a couple spaces after each comma.
Error bound of a Taylor series
Why It's Interesting
In Figure 6, are we all square with copyright?
-Diana 10:49, 15 July 2013 (EDT)
- "Click here to show the approximation of using Taylor series." : Using math environment shows π with a white box in the background, you can use "& pi ;" (get rid of the spaces here) to use inline representation. There might be other places in the text that in-line may look better, but I think the hide/show message would definitely look better with inline.
Throughout this section, I feel it would be clearer to use “nth order approximation,” if you’re talking about the Taylor polynomial of n terms, and “nth term in the Taylor series approximation” if you’re referring to just that term. But maybe what you have here is standard terminology that I just don’t know…
-Diana 10:49, 15 July 2013 (EDT)
Steve M 5/31/13
- At the bottom of the small angle section, before "how small is small", there are lines
- because for small x be only need the first-order approximations and the first-order approximation of cos(x) is 1.
- The other trigonometric functions may be obtained, as usual, by taking the reciprocal of sin(x), cos(x), or tan(x).
- Well, how do you take the reciprocal of a power series, or a Taylor polynomial? By long division. Perhaps this should be discussed.
How small is small?
- Do you think it's possible to make the expressions in the graphs as fractions? If you've tried and didn't think it looked good, ignore this then. Also, the blue curve in Figure 10a is very close to the x-axis therefore not very obvious. It might be better to adjust the scale a little bit.
- "If the approximation is useful, then they will find another" huh? isn't this "If the approx is NOT useful"?
- "any graphing software will plot the sine, cosine, and tangent functions almost to exactitude" huh? Is "to exactitude" exactly? Don't you mean that they can be plotted quite exactly? I don't follow.
- What are first-order and second-order approximations?
Removed a section on the use of the small-angle approximation in finding a formula for a pendulum in closed form. Diana noted that this was getting pretty far removed from Taylor series; the small-angle approximation is an application of Taylor series, and the pendulum formula is an application of the small-angle approximation. There is really one line in the derivation of the formula that pertains to Taylor series; the rest is physics and differential equations. So it has been removed, and I will add a much shorter explanatory paragraph in the small-angle approximation section in order to just mention it.
The text is left below in case it need be reinstated. There is a mention at the end of the text from the pendulum section below about the possibility of finding a more exact (although less pretty) approximation for the motion of a pendulum using composition/inversion of Taylor series (this would involve going to Eq. 6 and, rather than substituting in the first order approximation, you'd substitute in a larger approximation). This way, the pendulum section could actually be "salvaged" and brought back more thoroughly to the subject of Taylor series. I had talked to Steve about doing this, but the page became way too long to consider adding more, so it was never done. But it could make for an interesting future addition or future page. Possibly.
Part of the reason this was discussed is because some of the math on the Rope around the Earth page uses a similar method of finding a series approximation using the inversion of Taylor series. That page links to an unmade Series Approximations helper page. This Taylor series page is an image page and focuses on the "basics" of Taylor series, and the composition/inversion don't fit here, but there are possibilities for anyone interested in taking Taylor series further. Particularly, Steve and I had discussed the possibility of an "Advanced Taylor Series" page that includes things like inversion/composition and Taylor series in several variables, which could be a future option for anyone interested - although I'm not sure if there is any precedent on math images for such an "advanced" image page.
--Gbrown2 15:13, 15 July 2013 (EDT)
This entire section doesn’t really seem to belong on this page. It’s really fascinating, but not particularly related to Taylor series. I’d just take the whole thing out. If you want to keep the info around so others might develop it on another page later, you could put it on the discussion page or as a hidden section in ideas for the future.
-Diana 10:49, 15 July 2013 (EDT)
In the simplest respect, the small-angle approximation is "close enough," and it's quicker than evaluating more terms of a Taylor series. However, the small-angle approximation has an additional utility in that it can allow us to solve certain differential equations in closed form. Or rather, it allows us to find an exact closed-form solution to a simpler differential equation which is approximately correct, making the closed form an approximate solution to the exact differential equation. A particular example of this utility is the derivation of the closed-form approximate equation for a simple pendulum.
This explanation uses knowledge from physics and differential equations.
Figure 11 on the right is a diagram of a simple pendulum. The force due to gravity on the object is mg. Using basic trigonometry, this downward force can be decomposed into a force parallel to the string and a force perpendicular to the string. The force parallel to the string (mg sin θ) is canceled out by the tension in the string, so the only force acting on the bob is the force perpendicular to the string (mg cos θ). For this reason the direction of the object's instantaneous motion is perpendicular to the string.
We would like to find a general equation for θ(t), the angle formed by the string and the vertical as a function of time.
We begin by noting that
- , where s is linear position, so
Furthermore, we know
- where a is linear acceleration.
Since the only net force acting on the bob is the force due to gravity perpendicular to the string, we have
Substituting and simplifying, we get:
This is where the small-angle approximation comes in. Because of the expression, we cannot solve this differential equation in closed form. Instead, here we will substitute in our small-angle approximation for sine:
Differential equations of this form have solutions in terms of the sine and cosine functions. Consider, for instance:
- where k and c are arbitrary constants
In this case, is , so we obtain the solution:
By dividing each side by L, we may put the equation back in terms of θ.
is generally known beforehand; it is the largest angle formed by the bob in the pendulum's arc. The phase shift is found by finding a zero of . Any such solution for the differential equation could alternatively be expressed as a sine function with a different phase shift.
At this point it is important to reflect that Eq. 7, a relatively simple analytical solution, was made possible only by the assumption of a small angle. Naturally, this means that our formula does not work for larger angles. For larger t, or after several oscillations, the approximation will become gradually less accurate, since it is not an exact formula; the error will be compounded over time.
For future addition of composition example: How might we obtain an even more accurate formula for angular position using Taylor series? Returning to Eq. 6, we can actually find a more accurate solution using composition of Taylor series...
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