For calculus students, what should be the intuition or motivation behind series?

Question

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?

Answer 1

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.

Answer 2

(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.

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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?

Answer 3

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.

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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.

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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.

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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?

Answer 4

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:

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.)

Answer 5

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

Answer 6

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 $\sum_{k=1}^{\infty}9/(100^k)$

Answer 7

I start on the other end of the spectrum. I start off by

1. Noting that computers can't compute sine, cosine, or anything like it. Computers can basically only compute polynomials.
2. Sine and cosine can't be written in terms of polynomials
3. However, we can get around #2 if we use an infinite number of terms. I point out how the degree of polynomial tells you how many humps you have, and so, to get something with infinite humps (like sine/cosine) you certainly need polynomials.
4. Then, we go over simple series for basic functions with MacLauren and Taylor series.
5. Then, I show that we can compute impossible integrals using infinite series (like integral of cos(x^2))

So, at this point, they have a basic set of reasoning skills about infinite series. They know why they are important (it allows you to perform calculations using computers, and it allows you to do previously-impossible integrals). So, now they know the importance of series. Once they know the importance of series, they can learn their limitations. This is the point where you can teach about convergence and divergence. For instance, you can use the Taylor series for natural log about 1, and show, using information about the rules for series, why this series only succeeds for a very small domain.

I think the notion of starting with a bunch of abstract concepts about series is a severe turn-off. If you start with why we care about series, and start using series, you can then back up and talk about conditions required for the series to converge to an actual value. This keeps students attention much better than the standard approach.

Answer 8

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.

Answer 9

The fact of the matter is series is sort of viscerally off the beaten track of limits/derivatives/minmax apps/integral toolbag/integral apps. It's there because it is needed later (and covered more) in ODEs, PDEs, and (for math majors) real analysis. You definitely don't want to torture yourself (and the class) by spending huge time on justifying what is a bit of a side trip [even if someone comes on and says it's not a sidetrip, it sure FEELS like one, which is the important thing].

That doesn't mean you have to be totally just "do this". But a couple remarks to the effect of "needed in later classes" as well as the stereotypical "series can be used to approximate smooth functions" is enough. And yes, without proving, showing etc. those two comments. Then get on with it. Teach the topic and be energetic and follow the text. And test them. And move one. Don't make something that is a sidetrack even sidetrackier. Give them the exposure (as with baby determinants in HS or Laplace transforms in ODEs)...and then roll on and away.

Also, even though it doesn't feel as amazingly wonderful and applied and organically connected as minmax or integration by parts, it's not the worst thing in the world for the kids to do the mechanical work in fiddling with series. There's a fair amount of just paper and pencil symbol algebra in series and you see it from second year (US blabla) HS algebra to calculus to ODEs to PDEs to numerical analysis to real analysis to complex analysis. (Realizing everyone does not take all of those, but engineers will get a lot of that, even if in an "engine math" course.) I don't really remember the rules for convergence...I mean it was good to learn it once, and at least know intuitively that some things that sorta look like they oughta converge don't. But the "muscles" of writing terms and all that was good stuff. And in some ways similar to the "muscles" of dealing with equations for EE in circuit analysis or fluids hydraulic circuits or thermo power cycles.

Answer 10

I motivate series for their primary application in the course, representing functions as power series. And I motivate power series in a couple of ways. First, they are a vast generalization of polynomials whose differentiation and integration rules are basically as simple as the power rule for polynomials (plus the implicit linear structure of differentiation and integration). Never underestimate the value of something that can be easily computed! Second, Taylor series are refinements of the key idea of calculus -- linear approximation. They are used to approximate transcendental quantities using only the four basic arithmetic operations.

Answer 11

In terms of does a sum converge, I motivate by comparing growth rates of functions. So powers of n grow slower than exponential functions of n, which in turn grow slower than factorials (as n goes to infinity).

So

$$ \Sigma \frac{2^n}{n!}$$

will converge simply because the denominator has the faster growth rate. Using this idea, my students are able to determine if a series will converge just by looking at it. They then use a convergence test to confirm their intuition. They also know what to compare t, since they know to ignore slowly changing terms.

As a bonus they also learn about growth rates of functions which has a lot of applications.

Answer 12

-1. Ask what $D^{50} \left\{ \ x^{100} \ \right\}$ is. Stop them from actually multiplying the coefficients; just leave them unmultiplied the way you would for a prime decomposition. Define factorial and show how $\displaystyle \prod_{50}^{100} \bullet = {100! \over 50!}$ shows up naturally in this context of repeated derivatives. This proves that arbitrary number of $D$'s of a polynomial is easy.

1. By fundamental theorem of algebra, $\mathrm{Polynomials}$ cover all functions (so, school algebra can work in the real world)
2. By Taylor's approximation, $\mathrm{Finite\ Polynomials}$ can get you close (so, it actually could work, in the real-real world)
3. Each polynomial is "just a" sequence. (The covector of coefficients to the one basis-vector $(\ldots, x^{-1}, x^0, x^1, x^2, \ldots)$ (with $x^0$ moving the root to wherever you want

4. We can understand sequences. (There is an analogy to digits, possibly negative, in base $x$, but that could be ignored.) We have a shift / lag operator ($D$) on these sequences. (−1) So symbolic differentiation & integration of these letters / strings / symbols, gives us an "Easy way to do calculus" (No hard integrals! Only polynomials). If most functions can be written this way (1), and we can even know beforehand (in the planning phase) how good of a job this approach will do (2),

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Now play around with them — do ℂ→ℂ Wegert-plots and Cartesian ℝ→ℝ plots, and input/output tables to see how varying the "sliders" (entries in the covector of coefficients) changes the shape of the output in the various views (Cartesian, Wegert, tabular). A computer/interactive setup could be the best here. Just mess around and try to get the output to do cool stuff.

exp(ζ)/(ζζζ−ζζ−ζ+i)

{ ζζζ−ζζ−ζ+i } / { log(ζ)+i }

(For the ℂ→arg ℂ plots you want to tell them that arguments swirl around the roots ($\arg \zeta$ takes on every argument at 0)

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For applications, you can point to some interesting things to do with ℂ maps (dynamical systems, root finders, some of http://arminstraub.com/math/what-is-column).

Further applications you can discuss net present value, the geometric series (and the dumb geometric-series trick), the second dumb trick of subtracting two shifted geometric series to get a finite one (since no-one lives forever), and how although rates-of-return might vary in reality, we've now carved out some "easy cases" that we can do back-of-the-envelope, as quickly as $1 \over 1-r$ $- \mathrm{ shift} \left\{ 1 \over 1-r \right\}$ to put some easy bounds on real-world cases. (If your students don't already know how to compute a NPV in Excel, then here is your chance to teach them the only mathematics they will actually need to know in 90% of office jobs.)

You can also tell them they now understand (finally) what $e$ is. All the little kids are just made to use it, they can now explain it. (Identify it with the covector $(1,1,1,1,1,\ldots)$ once you adjust by $/ \sim k!$.) And show how $D(e^{\bullet}) = e^{\bullet}$ via $(1,1,1,1,\ldots) \overset{D}{\mapsto} (1,1,1,\ldots)$. Reinforce the shift-arithmetic on symbols p.o.v., mention why this makes $\exp$'s a good basis for ODE's, and mention the "slick" version of instantiating $\sqrt{-1}$ (simply assume it) --- versus seeing it in series. (So which is better? Chance to make a value judgement / form an opinion on mathematical beauty.)

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It's not a bad time either to talk about the history of $\sqrt{-1}$: it was only possible to discover this "algebraic" / "axiomatic" / "slick" thing (with so many implications) by using stuff from another part of mathematics (analysis, but let's call it series / string-ops). By staring at and doodling around (mention Ramanujan) with the $\exp$ series, only if you've also tried to approximate the circle height function ($\sin$) and its cousin by infinite-approximation (mention Aristotle)

To me $\sqrt{-1}$ comes from the intersection of different areas of mathematics, i.e. from different ways of thinking being tied together by writing down on paper. This is a thesis about the way mathematics works as well as about creativity, exploration, proof, dealing with your own ideas, how paper can help you organise your thoughts. (in other domains perhaps, as well)

Now the discussion has gone from meaningless symbol-scratching to high-level.

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The usual approach (Stewart) is, in my opinion, insane. I would show the Stewart chapters as enrichment. This is heavy symbol scratching with no motivation why it matters at all. Yet another exercise in making people do mechanical calculations, instead of teaching them anything about mathematics.

Series are here to make our lives easier. They either make the impossible possible (formal power series, Ramanujan stuff) , make hard integrals turn into shift-operators, or they give you some peace-of-mind that (pretty much) any ℝ→ℝ mapping the real world might throw at you, can be reduced to some stuff you could do on paper, in a symbolic differentiator, etc. Reduced to essentially a string / list.