# What good is the phrase “Taylor series”?

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?

• I have a similar dislike for the term "Maclaurin Series". I just say Taylor/power series centered at zero, but the book insists otherwise. – Aeryk Dec 13 '15 at 18:44
• @Aeryk Yeah, that terminology is even more ridiculous. – Tim kinsella Dec 13 '15 at 18:45
• I've never heard anyone refer to a Taylor series as a power series, so if you're going to throw out a redundant term, I'd suggest throwing out "power series." "Taylor series" is universally used, so students need to know it as preparation for the real world. – Ben Crowell Dec 14 '15 at 12:34
• Related Question on Math SE: Are Taylor series and power series the same “thing”? – R.M. Dec 14 '15 at 17:46
• Besides "power series representation" there is also the useful term "power series expansion," which I prefer. And I agree with you about preferring the term "power series" to "Taylor series." If I talk to another mathematician, for instance, I always use "power series." But like the need to write ln instead of log when teaching students (in the US, in my experience) in order to be consistent with their book and with notation they see in science courses, for me the term "Taylor series" exists only in teaching. I despise the term "Maclaurin series" for power series centered at $0$. (Contd.) – KCd Dec 16 '15 at 12:14

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.

"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.

• In fact it's a theorem of Borel that for any sequence of real numbers $a_n$, there exists a function $f: \mathbb{R}\rightarrow \mathbb{R}$ such that $f^{(n)}(0)=a_n$. So in fact every power series is a Taylor series. Even if this were not true, it would still be perfectly fine to speak about the power series of a function. But I think your point is a good one. At least it's made me think about analogues of this situation in everyday language. Thanks. – Tim kinsella Dec 14 '15 at 7:42
• If I had to raise an objection to your examples, it would be that in each of them the super set is much larger than the subset. In our case, in fact, the two are identical (due to Borel), but even if this weren't the case, we really only care about Power series that converge in some open neighborhood of their center. Those that don't are rather useless. So even without Borel, the compliment of the subset is comprised of useless power series, and in my opinion, such a set doesn't warrant the introduction of new terminology. – Tim kinsella Dec 14 '15 at 8:02
• It would be silly to insist that no one use the phrase "Labrador" precisely because the compliment of this set is enormous. Likewise for your other examples. In our case the compliment is empty (or, even without Borel, negligible). – Tim kinsella Dec 14 '15 at 8:04
• @Timkinsella Okay, I'll accept that for any given non-polynomial function, there is a unique "power series", and that that is the Taylor series. But the whole point of the Taylor series is not that it is simply the answer to a question, but that it is the result of a specific process: it embodies a method for finding the answer. I suppose that technically if you asked me to find the Taylor series for f(x)=x^2+7, I'd say that the answer is x^2+7. That's true but trivial. From a pedagogical point of view, "Taylor" really describes a method, and it's useful to give it a name. – Jay Dec 14 '15 at 14:36
• @Timkinsella RE "compliment is enormous" Sure, but the number of members in a set isn't really relevant to whether it's useful to give that set a name. Like I mentioned we talk about natural numbers, wholes, rationals, reals, and complex. All numbers are complex. (Or if there's a bigger set that I'm forgetting or am not familiar with, whatever that is.) Would you say that we shouldn't use the term "complex number" because the complement of that set is empty? I'd say it's a very useful term. If we just called them "numbers", it would be hard to explain how they're different from reals, etc. – Jay Dec 14 '15 at 14:40

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.

• In France, the term "polynôme de Taylor" does not include the remainder term $o(x_x_0)^n)$, unlike the term "développement limité". In any case, I do not really pretend to solve your issue, which seems to have deep roots in US (bad) terminology. – Benoît Kloeckner Dec 14 '15 at 9:18

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.

• This is also my viewpoint. Given a function $f$ which is smooth at a point we can generate its Taylor series centered at that point. The function is analytic at that point if the generated Taylor series and function are coincident on some nbhd of the given point. In contrast, a power series is a function whose formula is given pointwise by a power series. So, yes, the distinction is the starting point. – James S. Cook Jun 26 '19 at 15:50
• Most often today calculus is taught in the "early transcendentals" way, so we should always be speaking of Taylor series of various functions that they already "know". At the very end of the course, one could mention that none of those functions was rigorously defined, and to "rebuild" everything from scratch we should adopt the view that functions are defined by power series. This way of redefining something by its "derivatives" [in the sense in everyday English] is what mathematicians constantly do, and we call it the abstract or axiomatic approach. – liuyao Jan 20 at 19:25

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.

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.

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.

• Fourier series are relevant in an abstract generality which doesn't have much to do with trig functions. As far as I'm aware (not very!) nothing like this is true of Taylor series. With respect to your remark about Laurent series, clearly we want a medium between terminological proliferation and terminological poverty. The question is how useful is the distinction between the concepts we choose to denote differently. The context in which one encounters power series is pretty big, and in those contexts, the fact that the series does not have terms of negative degree is very important. – Tim kinsella Dec 14 '15 at 2:54
• Constantly writing "Laurent series with no terms of negative degree" would be pretty exhausting and inelegant. It would result in more clutter than the introduction of new terminology.But in cases of a distinction without a difference, the terminology only serves to clutter our perspective. – Tim kinsella Dec 14 '15 at 2:54