What good is the phrase "Taylor series"?

Question

I've taught integral calculus a few times, and in every course the students are confused about the distinction between Taylor series and power series. It's something I remember being confused by as a student. Now that I have a little more background, it seems to me that there's no difference at all, and that one could and should replace the phrase "Taylor series of" with "power series representation of." Of course non-analytic functions won't be equal to their Taylor series in any neighborhood of the center, so calling it a "representation" of the function might be a bit misleading. But most or all functions we meet in a basic integral calculus course are analytic (I'd wager all $C^\infty$ functions you show your class are analytic, unless you cook up one of the standard counter-examples for that purpose), and it would be easy to simply state this caveat for non-analytic function. In fact, you could dispense with the word "representation" and just talk about the "power series of f(x) with center a".

So are there good reasons to keep the confusing terminology which seems to imply a distinction without a difference?

Answer 1

Taylor series are an extension of the concept of a Taylor polynomial. A Taylor polynomial is a polynomial which approximates a function, and is constructed from the derivatives of the function at a specific point. It is reasonable to think that there would be many different polynomials that might approximate a particular function, and the Taylor polynomials are the ones constructed via a certain process using derivatives. When you take your polynomial to infinite degree, you get a Taylor series. Calling it a Taylor series preserves this connection to Taylor polynomials and derivatives.

Consider the statement "every power series that converges in an interval is a power series representation of some function" -- well duh! If it converges then it has a function value. Compare to the statement "every power series that converges in an interval is a Taylor series for some function" -- this brings to mind the idea of the derivatives and the $n!$, which is more meaningful. Moreover, it somehow creates more of an expectation that you might sometimes be able to guess the name of the function. Something like "Oh! It's the Taylor series for $\ln(x)$ centred at $x=1$!".

Also, regardless of whether you use the terminology in your local classroom, you do still need to tell students about the other terminology, so that they can read and use other people's resources. (It's an interesting exercise to google "Taylor series" and "Power series representation" separately.)

My personal preference is to use "power series representation" most of the time, but sometimes say Taylor series too, mostly when I want to make the point about the derivatives, but sometimes just to mix it up.

Answer 2

"Power series" is a very general term. I'm not a mathematician, but as I understand the terminology, the term "power series" refers to any polynomial, not just one derived as a Taylor series. Yes, all Taylor series are power series, but not all power series are Taylor series. And the process of generating a Taylor series is very specific and important. It makes sense to give it a name.

In a way, your question is like asking, "Why should we have the term 'Labrador retriever'? It's just a dog, isn't it? Why don't we just call it a dog?" Yes, we could, and in some contexts that would be fine. But sometimes we want to be more specific.

Even if the student doesn't remember all the names, I think it is helpful when learning to have specific names for specific things. Giving something a name makes it easier to talk about it, and I think aids comprehension.

Like, in math we often talk about the difference between integers, rationals, reals, and so on. Why don't we just say "numbers" and leave it at that? They're all numbers. But we want to make the point that specific kinds of numbers have specific properties. By giving them names we make it easier to talk about them.

Answer 3

In French we have three phrases which are clear and relevant to different situations:

• "développement limité" which is translated as "Taylor series" and means a relation of the form $$f(x) = \sum_{k=0}^n a_k(x-x_0)^k +o(x^n),$$

• "Théorème de Taylor-Young" (easy to translate) which is the statement that a function which is derivable at $x_0$ has a "développement limité" up to its derivability order at the considered point, with the coefficient given by the normalized derivatives at $x_0$,

• "séries entière" which is, I think, translated by "power series" and means an expression of the form $$\sum_{k=0}^\infty a_k z^k$$ in a formal sense, i.e. independently from the convergence radius of the series and with $z$ a placeholder.

I think this gives a precise and distinct role to each part of the theory. In particular, a Taylor series has finite order and is a mean to approximate a function, while power series representation is an exact equality with an infinite sum.

Answer 4

Even though "Taylor series" and "power series" are the same thing, the names suggest different viewpoints. When I hear "Taylor series", I think that there was first a function $f$ under consideration and then a series was obtained by evaluating $f$ and the derivatives $f',f'',\dots$ at some point, etc. When I hear "power series", I get no such preconception of where the series came from. It might have been obtained as the Taylor series of some function, but it might also have arisen as the generating function for some combinatorially defined sequence of numbers, or it might have arisen in some totally different way.

Answer 5

I think the existing answers, at this time, all raise important points, and I have nothing to add to the ones already made. However, I feel it is worth emphasizing that, when teaching, it is important to bear in mind the ultimate purpose of the education, and not “make the material fit our teaching” but “make our teaching fit the material.”

There is simply no question that, if you are teaching students who are studying (or will go on to study) physics, engineering, math or any other subject that uses applied mathematics, they would be quite literally “uneducated” if they did not understand the difference (or lack thereof) between “Power Series,” “Taylor Series” and “Maclaurin Series.” It is simply part of the language and culture of these fields, even if confusing when first learning, and/or inconvenient when teaching.

Of course, if the students will not go on to use these tools and the content is covered in a course that marks the end of their studies, then it is surely less important (for the reasons covered above).

A related issue is the language used to express the centre of convergence: “centred at $x = a$” and “in powers of $(x-a)$ are both equivalent expressions that, for better or worse, students who will use these ideas in the future need to know.

In short, the language really does matter for some students; however, the distinctions are easily introduced once a firm understanding of the ideas exists (much as the difference between different breeds of dogs is easily introduced once one has a firm grasp of “dogs” as a group, first!); in fact, confusion in the terminology can be used to diagnose a lack of understanding of the basic ideas and can, in that way, be turned into a pedagogical advantage.

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On the practical considerations associated with teaching these topics:

I have found the most success (in avoiding confusion surrounding the language used) by introducing the idea of a power series, developing the related idea of power-series representations, and then simply mention, as an historical / cultural note, that “Taylor/Maclaurin series” is commonly used instead of “power series representation,” once students are comfortable with the ideas. Similarly, I have found that students are better equipped to understand the connection with “tangent lines” and “linear approximation,” after they have grown comfortable with the definitions and computations involved in power series (representations).

I’m not sure if the phenomenon is widespread, but students in my neck of the woods generally seem to have a very poor understanding of power series and power series representations (reflecting an equally poor understanding of tangent lines and linear approximation), in general, so focusing on clearing up these points usually makes the later clarification of terminology a non-issue.

As suggested by others in the comments and other answers, I have found that emphasizing power-series expansions as an extension of the tangent line is fundamental to helping students understand (again, from an applied-maths perspective, I would want all my engineering and science students to know that “first-order Taylor series expansion,” “tangent line” and “(best) linear approximation” are all equivalent expressions).

Finally, a strong emphasis of graphical presentation of the material always seems to be essential for most students.

Answer 6

Indeed we could get by without referring to Taylor series, and instead just always refer to a "power series." But then maybe we should also stop referring to Fourier series and simply refer to "trigonometric series."

I think that "Fourier series" draws attention to the particular formula that relates the coefficients to a function that is being represented or approximated. The same is true for "Taylor series."

As to the redundancy of "Maclaurin series" since they are a special case of Taylor series, we could take this a step further and forever replace "Taylor" with "Laurent" since Laurent series are more general.

I think mathematics is culturally enriched by references to constructs of Taylor series, Maclaurin series, Laurent series, Fourier series, Puiseux series, etc.