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I've noticed that series are one of the most difficult portions of calculus for new students to learn.

I think the level of abstraction has to do with this. Limits, derivatives, and integrals, as well as continuous functions, all have helpful visual meanings that give students good intuition (derivatives are slopes, integrals are area, etc.)

This intuition helps them remember certain facts. Why is the derivative of a constant zero? Because the slope of a horizontal line is zero. Why is $\int_a^a f(x)\, dx=0$? Because a line segment has zero area.

There seems to be no similar intuition in my calculus courses. If I ask, "Does $\Sigma_{n=0}^\infty 1$ converge or diverge?", many students have to use some test and often get it wrong.

Even more, to the students, there seems to be no motivation behind series. Derivatives are velocities, and integrals give area and volume, but series don't seem to many students to have similar uses. This compounds the intuition problem.

What visual intuiton or real-life motivation will help students understand series the same way that slopes or velocities tell students understand derivatives?

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One approach I'm trying is to emphasize the analogy between infinite series and improper integrals, which in my class we spent a while on earlier in the semester. Even though infinite areas are counterintuitive in some ways, at least there you can "see" the region and think about its area. But this isn't quite direct visual intuition or real-life motivation. –  Mike Shulman Apr 2 at 21:09
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@MikeShulman: I'm doing that as well. I'm not sure if it's helping as much as I anticipated, though. It's still unbelievable to students that there can be finite area under a curve that "goes to infinity". –  brendansullivan07 Apr 2 at 21:19
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Some brief observations, thus more of a comment than an answer: point out that all decimal expansions are really examples of series, do a lot of work at the beginning with geometric series (pays off later, for example in seeing why the ratio test works), introduce quadratic approximations (helps motivate Taylor series later) after linear approx. back at the beginning of calculus (or include it early in the calculus 2 course as a kind of review of calculus 1 stuff, if you didn't teach them calculus 1), use technology to look at graphs of $y = f(x)$ minus linear and quadratic approximations. –  Dave L Renfro Apr 11 at 13:56

7 Answers 7

up vote 12 down vote accepted

In my experience, one of the problems with series is that usually you have two sequences if you investigate the series $\sum(a_n)$: the sequence $(a_n)$, and the sequence of partial sums $S_n=a_1+\ldots + a_n$. I noticed that trying to stress this distinction helps a lot.

To the intuition, I like R. Péter: Playing with Infinity, the chocolate bar example on page 105.

A well-known mathematician (while still a child) interpreted the meaning of the sum of an infinite geometric series in the following way.

There was a type of chocolate which was popularized by putting a coupon in the wrapping, and anyone who could produce 10 such coupons would get another bar of chocolate in exchange. If we have such a bar of chocolate, what is it really worth? Of course it is worth more than just one bar of chocolate since there is a coupon in it, and for each coupon you can get $1/10$ of a bar of chocolate. But with this $1/10$ of a bar will go one tenth of a coupon, and if for one coupon we get $1/10$ of a bar of chocolate, for $1/10$ of a coupon we get $1/100$ of a bar of chocolate. To this $1/100$ of a bar of chocolate belongs $1/100$ of a coupon, and for this we again get $1/10$ as much chocolate, i.e. $1/1000$ of a bar of chocolate, and so on... This process can go infinitely, so that my one bar of chocolate together with its coupon is in fact worth $$1+ 1/10 +1/100 + 1/1000+...$$ bars of chocolate. On the other hand, this is exactly 10/9 of a bar of chocolate. 1 is the value of the actual chocolate and the coupon that goes with it is 1/9 of a bar of chocolate. Indeed 9 coupons are worth one bar of chocolate because you could ask for such a bar and tell that you will pay after you eat it. So you eat it, take out the accompanying coupon, and give 10 coupons back which is exactly what is needed for one chocolate bar.

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Regarding your first point, about stressing the distinction between a series and its sequence of partial sums, this recent thread is highly relevant: matheducators.stackexchange.com/questions/925/… –  brendansullivan07 Apr 3 at 18:47

Another point that oughtn't be neglected was that, historically, numerical computation/approximation used Taylor-Maclaurin expansions (not to mention Newton-Raphson when convenient) to approximate root-taking. Newton was apparently very happy with his discovery of the binomial expansion for general exponents, although surely not only for numerical purposes.

For historical verity, though, one perhaps should mention Poincare's discovery that some of the venerable series used in astronomical calculations were really only asymptotic expansions, so gave good-up-to-a-point approximations, but were actually divergent. This phenomenon is already visible (once one knows to look) with irregular singular points of second-order O.D.E.'s

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Unless the venerable series come up on their own, I'd leave this out (for now). Better not confuse their tender minds... –  vonbrand Apr 3 at 1:48
    
@vonbrand, sorry I didn't notice your comment earlier. I would disagree, though not that one/we/I/you should "punish" students for "failure to appreciate". Only that there is considerable virtue in "letting on" the reality, even if we don't "have it on the final exam". –  paul garrett Apr 12 at 23:45

Unfortunately, my example is not a full answer to your question, but I think, it helps students to be at least beware of what can happen.

You can explain series as a inifite summation of areas (at least as long as everything is non-negative). And convergence of a series means: Is there a big rectangle such that all the given areas fit in? Intuitively, most students would say in general: That is not possible since there is always coming more to the sum. But a simple example of the geometric series can be given in a picture:

geometric series

You can also take some more sophisticated picture like the Koch snowflake and ask about the area. (This example illustrates the concept of partial sum: At some step n of the snowflake, the given area is the partial sum while in the series you add up the area of all triangles of a given size.)

Koch snowflake

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I think the Sierpinski triangle is another good example to use and might even be (slightly) better since the "border" of the figure is fixed, and the area at any iteration is obviously a sum of triangular areas. –  brendansullivan07 Apr 2 at 21:14

(I have not done this exact presentation, so I cannot vouch for its efficacy. But I have used the main idea before, and it seems to help some students, and is at least a bit of fun. Also, this is meant to address the intuition aspect of the question, not the motivation.)

Demonstrate by walking! Convergence means there's a spot that will be approached to any desired level of proximity, given infinite time. Divergence means a journey for which we cannot identify one specific destination.

Write the series $1+\frac12+\frac14+\frac18+\frac{1}{16}+\cdots$ on the board. Start at one wall of the classroom and take one giant step forward. Announce this to be your standard measurement of one step. Get a volunteer student (let's call him Joe) to go stand at the spot that is 1 "step unit" away from where you are now. Announce that Joe is your destination.

Next, take a step forward that is half as big as your first one. Ask the students how many "step units" you have traveled so far. Get another volunteer (Sam) to write this on the board. Take a $\frac14$ step forward. Ask them how many "step units" you have gone so far. Sam writes this down. Keep doing this, having Sam record this sequence of partial sums (you can even call it that now). After several iterations, they'll see that you're standing right in front of Joe. Ask them what will happen if you keep going.

Announce: "This series converges to 2. If I were to keep taking paces like this forever, I would never be able to step beyond Joe. I will never actually be standing exactly where Joe is, but if you want me to get within a distance $d>0$ of Joe, I can definitely get there, no matter how small you make $d$. But the smaller $d$ you choose, the longer you'll have to wait. Mathematically, the sequence of partial sums approaches 2." (At this point, you can even show them the pattern that the partial sums obey: $s_n=2-\frac{1}{2^n}$. Ask them for $\lim_{n\to\infty} s_n$.)

Go back and write this series on the board: $1+1+1+1+\cdots$. (I chose this based on your example in the question.) Start walking by taking paces of size "1 step unit", having Sam record your sequence of partial sums. Walk straight out the door. Ask the students loudly from the hallway, "What is my destination now?"

Come back and announce: "This series diverges. The sequence of partial sums [point to $s_n=n$] grows without bound. This is an infinite journey that will not approach a specific place. I cannot tell Joe where to go and wait for me because, no matter where he stands, I will eventually walk straight past him."

I think this will be a striking presentation and will at least demonstrate to students the different natures of con/divergence. You could even do this with other series (and later on, with any series that comes up, you can remind them of this discussion by just starting to take paces across the room).

Take, for example, $1-1+1-1+1-1+\cdots$ and show that you just jump back and forth by 1 step and never settle on a specific place for Joe to wait for you. Then, take $1-\frac12+\frac14-\frac18+\cdots$ and tell Joe to wait at the spot corresponding to $\frac23$. Show that you'll step back and forth around him, getting closer and closer all the while. (Refer back to this when you do the "alternating series test".) You could even bring in Sam for this part and show what the sequences of partial sums are doing.


As for the motivation behind series, I like to mention the idea of Taylor series right away, and state that they are both very powerful and very common, and can be used to approximate functions accurately and efficiently. I don't say anything yet about how to find them, but I try to give the students an idea of where/how/why they're useful.

For example, I point out that the approach to many physics problems (at least in the courses they'll take) is "Write the Taylor series and chop it off at $n=2$ (or even $n=1$)."

Also, I ask them what they think their calculator is doing when they type in $\sin(0.35)$, say. How does it know what that number is? It's not like it has a big list of all the values of the sine function. There's infinitely-many! How does it spit out an answer?

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Going beyond: If you are walking in steps of size $\frac{1}{n}$, then you will "walk off to infinity" (since you have the harmonic series). If you take steps of this size but rotate 180 degrees after each one, then you will converge to some point (since you have the alternating harmonic series). If you rotate 90 degrees after each step, then you will be walking in a sequence of nested squares, where the area of the squares becomes arbitrarily small; so you will converge to some spot (by a "nested square" property). More generally, see this MO post: mathoverflow.net/q/109582/22971 –  Benjamin Dickman Apr 2 at 23:00

It seems to me that the reason for learning series in calculus is to analyze functions through power series representations. There are other reasons series are important in mathematics, but in the standard calculus course, it seems to me that series are introduced for the sake of explaining power series. The key is then,

Can one motivate power series before even covering series?

I think so. To some degree, you have to sell it, and then deliver later in the course. I tell the students that series have been important in understanding difficult functions, such as algebraic and transcendental functions. I sometimes make rather broad claims that are only roughly correct. For instance, when you press a button on a calculator, how do the engineers know that the digits are correct? (I guess I often motivate something by asking questions that I don't answer; sometimes that I can't answer.) I also tell them they will be able to see how a function is related to all its derivatives and see how L'Hôpital's rule, montonicity, and concavity work. @paul garrett has mentioned Newton's binomial series. One can also do some calculations that suggest the development of series.


For instance, given a small number $x$ with $|x|$ much smaller than $1$, approximate $\sqrt{1+x}$ with a linear polyomial $1+px$. (Pick $x = 1/10$ or $1/100$ for concreteness, and plug in to the results and compare with a calculator value as you go along.) So we would like to pick $p$ such that $(1+px)^2\approx 1+x$. If we match coefficients in order from lower to higher degree (i.e., from greater to smaller terms in order of magnitude), we get $p = 1/2$, with the left over term $p x^2$ being the error of the approximation; but $px^2$ will be much smaller than the linear term $2 p x = x$. Next improve the approximation by adding a quadratic term: $$ \left(1 + {1\over2}\,x+q\,x^2 \right)^2 \approx 1+x$$ Expanding and matching coefficients yields $q = -1/8$. And so on. One can see that this can be continued indefinitely. Is there an easy way to do it? Does it always get closer? [Since one probably does not care whether students can carry out these calculations, they could be written out on notes for the students to follow or done on a computer.] Alternatively, one might do the algebra all at once: Expanding $$ \left(1 + p\,x + q\,x^2 + r\,x^3 + s\,x^4 \right)^2 \approx 1+x$$ yields $$p={{1}\over{2}},\ q=-{{p^2}\over{2}},\ r=-p q,\ s={{1}\over{2}} \left(-2 p r-q^2\right)\,.$$ The solution can be completed through substitution.


Another example is that the repeated integration from $0$ to $x>0$ of $\sin x \le 1$ or $e^{-x} \le 1$ yields inequalities that show that $\sin x$, $\cos x$, and $e^{-x}$ lie between successive Taylor polynomials. (I would omit the word Taylor in class.) Since $n!$ grows so fast, you can approximate sine and cosine rather well over the first quadrant. Again it leads to questions about the indefinite continuation. This example can be done in Calculus I, if you want to give the students advertisement for Calculus II.


One question that motivates series, not dealing with power series, is the following: If you could stack as many books as you wish, how far out over the edge of the book on the bottom could the book at the top extend without the books falling over?

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I agree with your main point. There is a natural way to derive Taylor polynomials based on recursive use of the fundamental theorem, outlined in an answer to this question. The recursion makes it natural to wonder what happens if it's carried out indefinitely --- hopefully this motivates the notion of infinite series. –  Bob Pego Apr 7 at 2:51

Decimal versions of fractions give very simple examples. For example 1/3 = .333333333…..

And what is that decimal but $\sum_{k=1}^{\infty}3/(10^k)$

Or 1/11 is .09090909... Which is $\sum_{k=1}^{\infty}9/(100^k)$

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Perhaps you can use infinite series to expose your students to the notion that math can be beautiful with no practical applications.

If you add up an infinite number of terms, you would think you'll just get infinity. But this is not the case - as the simple example of $\sum 1/2^n$ shows, some infinite sums converge. This raises what I think is an inherently interesting question: which sums converge, and which ones diverge? To convince them that this question really is interesting, show them the harmonic series. Thought a series would converge as long as its terms tend to zero? Guess again.

Show them the fact that when subtraction gets involved, you can't always rearrange the terms in the series anymore. Treat the whole subject as a trip to the zoo: get them interested in all the odd phenomona that can happen when you add an infinite number of terms, and in how unpredictable the convergence or divergence of a particular series can be. The sum of the reciprocals of the integers diverges. But the sum of the reciprocals of square integers converges. What about the sum of the reciprocals of an arbitrary sequence of integers? Of the primes, for instance?

Of course, this type of approach may put an apparant gap between series and the rest of the material in the course, which might be presented as being much more practical.

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