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We introduce infinite sequences and series very thoroughly in calculus classes. We first define infinite sequences, then series, carefully discussing notions of convergence, etc., and discuss all sorts of rules for convergence before allowing students to see Taylor's theorem.

However, suppose that one just went to the board and wrote down $$e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots$$ without making a general definition of an infinite series, or explaining anything about convergence. (Presumably one would have to explain factorials so that the pattern is clear.)

Or, more simply, one could write $$1 = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$$

I have had interesting debates with colleagues as to whether this is a good idea -- and our debates seem to rely on an empirical question.

Are these formulas easily comprehensible to, say, Calc I students, bright high school students taking competitions, or other students who have not had formal exposure to infinite series? Or are infinite series a genuine conceptual stumbling block to students?

For example, would students be able to see how the first formula allows them to quickly find accurate approximations for $e$?

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Yeah, when we write $\frac{1}3=0.333...$, we are implicitly talking about infinite series. Although "1=0.9999...." still yields problems for young people who aren't clear on the formal nature of what the sum "actually" means. –  Thomas Andrews Dec 12 '12 at 14:20
The problem is, if you wrote $f(x)=1+x + 2!x^2+...$ your students would have no idea that that never converged. And if you wrote $1+x+x^2+...$ you would have to be specific that this only converges if $|x|<1$. –  Thomas Andrews Dec 12 '12 at 14:36
I would say that students have a little more intuition about infinite sequences and convergence. That said, it can be difficult to transfer this intuition to infinite series. –  Baby Dragon Dec 12 '12 at 14:40
Not what you're asking, but related: "Motivating Infinite Series‌​" –  Mike Spivey Dec 12 '12 at 16:45

2 Answers 2

For what it is worth: When I start teaching about series in calculus I usually start out asking them what one would get if one added $1$ to itself an infinite number of times. That is, I ask about $$ 1 + 1 + 1 + \dots $$ It actually doesn't take long for someone to answer that this would be infinity. I then repeat back to them that it is correct and that would make sense since you are adding an infinite number of positive numbers. I then get them convinced that this is the case: an infinite sum of positive number should always to infinity, right?

Then I write down $$ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots. $$ Now the whole class will call out that this is also infinity. Then I act all surprised. Then I draw a circle, and take out first half the circle, then a quarter and so on. Then I ask the question again: So what is this equal to? Again, it doesn't take long for someone to say $1$.

In my experience, you can introduce pretty complicated topics with a simple example to make it easier to digest initially. And it is also interesting to see how the students react when they realize that what I had just convinced them of $2$ min ago (which was so intuitive) was actually wrong. If you can have the students walk out of a class like that talking about how that philosophically makes sense, then I think that you have achieved something.

So, in my opinion, I think that you can introduce series pretty early on. If you want to have them prove stuff about series, then you of course need to talk about sequences and limits first, but I don't think that that series are in them selves a "conceptual stumbling block to students".

Again, in my opinion, you can bring a lot of math down so that it can be introduced to even high school students. It all really depends on how much you want to do.

I would have any concerns with giving an hour talk about series to a group of interested high school students.

Maybe the important thing is that you don't just on the series for $e^x$ as the first thing, but maybe wait until the students feel a bit comfortable with series. After that, you can simply give them a series and show them how series actually can be used for approximating things.

Then after that, show them this series:


Ask them if they can guess what the series is equal to (you might convince them first that it is not infinity). With a bit of acting, you can make it very surprising that this actually equals:

$$ \frac{\pi^2}{6}. $$

And them tell them that later in calculus they will get to see a proof of this.

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+1 for the mind-blowing act of showing the last equality just at the end of the lecture and then leaving them stupid ;) –  sonystarmap Dec 12 '12 at 14:46

Interesting question. In my personal opinion (almost done with B.Sc. in some math related subject, done teaching and exercises for lower semester students) I would say that this are two types of sums. Call me crazy, but it does make a difference, if you have $x$ written in that sum or not. \begin{align} 1=1/2+1/4+1/8+1/16+... \end{align} is something very intuitive. You can draw it, or try it with a calculator and even though you don't know that you can prove convergence, you get the feeling, that for more terms you get closer to $1$. An average smart math student would very soon come up with \begin{align} 1= (1/2)^1 +(1/2)^2+(1/2)^3+(1/2)^4+... \end{align} and if you then introduce them to the notation with $\sum$ they will understand, that the above can be rewritten as \begin{align} 1=\sum_{i=1}^{\infty} (1/2)^i \end{align} The second example however, is less intuitive. "What does an exponential function have to do with that polynomial?" But computer can close this gab.

Example: The teacher states that you can write $\exp(x)$ as \begin{align} \exp(x)= 1+x+x^2/2+x^3/6+x^4/24+... \end{align} an average student (or lets say I) would not immediately believe that, see question above. But you simply use some software to prove it. Make a video, show how your approximation gets better. The student will then maybe believe it. Even though one may doubt that this also holds for very large $x$. You then ask the students to find the pattern, which is the important part. How can you write these terms similar to the $\sum 1/2^i$ term above. \begin{align} \exp(x)= 1+x+x^2/2+x^3/6+x^4/24+... = \sum_{i=0}^{\infty} x^0\cdot ? \end{align} Therefore you need to tell them that $x^0=1$ and what something about $i!$. Again an average student will say: "Luck! Works only this time." And you repeat the above procedure with $\sin(x)$ or $\cos(x)$ or $\log(x)$. They will then see, that the sum is always the same, a.k.a. Taylor series. Then you state that this IS indeed always possible and to proof this there are some convergence methods. Of course you now have to start explaining these methods with a simple example like the one above, but from my experience, you can move a lot faster to more complex methods of convergence to also proof Taylor series. One is simply less intimidated by that horrific Taylor term.

So my approach would be \begin{align} \text{finite series } &\rightarrow\text{infinite series } \rightarrow \text{Taylor series } \\ &\rightarrow \text{simple convergence methods } \rightarrow \text{difficult convergence methods } \end{align} instead of \begin{align} \text{finite series } &\rightarrow\text{infinite series } \rightarrow \text{simple convergence methods }\\ & \rightarrow \text{difficult convergence methods } \rightarrow \text{Taylor series } \end{align}

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