General Feedback

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 $\lim_{x\to a} x^2$. 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.

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)

Basic Description

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)

Chengying 6/12/13

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

Finding the Taylor series for a specific function

Chengying 7/6/13:

"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

Chengying 7/6/13:

"Euler's number" link -> "Euler's Number"
"For instance, we can compute the Taylor series of the function composition $sin(2x^2)$": use inline format for sin(2x) and 2x2

Gene 6/11/13

"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.

Why It's Interesting

In Figure 6, are we all square with copyright?

-Diana 10:49, 15 July 2013 (EDT)

Approximating π

Chengying 7/6/13

"Click here to show the approximation of $\pi$ 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.

Small-angle approximation

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?

Chengying 6/12/13

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.

Gene 6/11/13

"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?

Pendulum

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.

--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)

Removed Text

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
Free-body diagram of a simple pendulum with mass m.

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

$s = L \cdot \theta$, where s is linear position, so
$\theta = {s \over L}$.

Furthermore, we know

$F = m a = m {d^2 s \over dt^2}$ 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

$- mg \sin {s \over L} = m {d^2 s \over dt^2}$

Substituting and simplifying, we get:

$0 = m {d^2 s \over dt^2} + mg \sin {s \over L}$
Eq. 6         $0 = {d^2 s \over dt^2} + g \sin {s \over L}$

This is where the small-angle approximation comes in. Because of the $\sin{s \over L}$ expression, we cannot solve this differential equation in closed form. Instead, here we will substitute in our small-angle approximation for sine:

$0 = {d^2 s \over dt^2} + g {s \over L} = {d^2 s \over dt^2} + {g \over L}s$

Differential equations of this form have solutions in terms of the sine and cosine functions. Consider, for instance:

${d^2 \cos (kt + c) \over dt^2} = -k^2\cos (kt + c)$ where k and c are arbitrary constants

In this case, $k^2$ is $g \over L$, so we obtain the solution:

$s (t)= A \cos(\sqrt{g \over L}t + B)$.

By dividing each side by L, we may put the equation back in terms of θ.

Eq. 7         $\theta (t) = \theta_{max} \cos (\sqrt{g \over L}t + \phi)$

$\theta _{max}$ is generally known beforehand; it is the largest angle formed by the bob in the pendulum's arc. The phase shift $\phi$ is found by finding a zero of $\theta (t)$. 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 $\theta$ using Taylor series? Returning to Eq. 6, we can actually find a more accurate solution using composition of Taylor series...

Feedback from Previous Years

Gene 7/15/12

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 12/14/09 ==== 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.