Why are infinite series so hard to grasp?

I’ve posted on infinite series a few times before. But I was inspired to touch on the topic again because I saw this post, yesterday, over at the Math Less Traveled. Actually, the post isn’t really about infinite series as much as it is about p-adic numbers and zero divisors. I’m excited to read more from Brent on this subject. But I digress.

The point I want to make with this post is that students struggle with wrapping their minds around convergent infinite series, and yet they live with them all the time. Students have inconsistently held beliefs about infinite sums.

The simplest convergent series is a geometric series \sum_{n=1}^\infty a_n=a_1r^{n-1} which converges to \frac{a_1}{1-r}. The easy proof of this fact goes like this: we look at the sum formula for a finite geometric series, s_n=\frac{a_1(1-r^n)}{1-r} and we notice that


for |r|<1.

But this proof isn’t very satisfying for the student encountering infinite series for the first time ever. Evaluating the limit feels like ‘magic.’ The idea of adding up an infinite amount of things and getting a finite value is unsettling. I admit, it sounds like quite a lot to swallow. That being said, however, students have no problem declaring the infinite series

0.3 + 0.03 + 0.003 + 0.0003 + \cdots

to be 1/3. It’s not “close to” 1/3, it’s not “approaching” 1/3, it IS EQUAL TO 1/3. And my Precalculus students already accept this as fact. So without even thinking about it, they’ve been living with convergent infinite series all along. Hah!

Once they finally shake their denial, they can more easily accept the convergence of other infinite series like \sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}. At first when students encounter a series like this, they think, “surely we can’t say the sum is EQUAL to \frac{\pi^2}{6}. It must be close to \frac{\pi^2}{6} or approach it, but equal to?” But the same students make no such distinction with 0.3+0.03+0.003+\cdots = \frac{1}{3}.

So there it is. An inconsistently held belief about infinite sums. To the student: You cannot have it both ways. Either you must agree with, or deny, both of the following equations:

0.3+0.03+0.003+\cdots = \sum_{n=1}^\infty 0.3(0.1)^{n-1}=\frac{1}{3}

1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots = \sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}

But to believe one equation is true and the other is only ‘kind of’ true is inconsistent. I rest my case. 🙂


8 thoughts on “Why are infinite series so hard to grasp?

  1. Thanks for the interesting insight. In my experience it seems that many students have simply accepted 0.333…. = 1/3 as a fact delivered from above, and have never made the connection to infinite series. But this certainly does suggest a nice starting point, pedagogically speaking, for talking about them!

    Also, just wanted to point out that I think you mean pi^2/6, not pi/6. =)

    Glad you’re enjoying my series on p-adic numbers and zero divisors, I hope to write more soon!

  2. IMHO, a psychological issue is at stake. 0.33333… = 1/3 is easier to swallow than 0.99999… = 1 because 1 is not a fraction, but 0.9999… clearly is a fraction. Or to put it contrapositively, why then is it not obvious that 0.01001000100001000001 … is not a fraction at all but 0.010101010101… is? Aren’t they “very similar” in their structure of alternating 0’s and 1’s?

    You might think that I get paid to say the following, and I do, because I teach History of Mathematics, but “ontogeny recapitulates phylogeny” in math as well as in fetal development. How did Aristotle handle infinite series? Cautiously. Zeno? Paradoxically. Euler? Formally. Cantor? Surprisingly. So surprisingly in fact that he adopted a philosophy of mathematics that might be called “psychologism.” (If you’re a mathematical historian, I’m referring to the “late Cantor.” The early Cantor might be viewed as having a theological view of infinities.)

    There. I’ve said it. Infinities. How can a concept be clear if it turns out that it is multiple concepts?

    Intuition is trainable. Teaching math at the very least includes training intuitions.

    • Great thoughts, Dr. Chase! 🙂

      Perhaps I should have included some of your good examples, which add clarity to the discussion.

      Would you agree that my last statement is true? That is, do you agree that you either have to accept both equations or deny both?

    • Since we’re on the subject of correcting insufficient understanding of the scientific or mathematical past, it’s worth noting that although it was once believed that ontogeny recapitulates phylogeny in fetal development, this is now known to be false. It’s still a clever analogy with math of course.

  3. I disagree, because there is an undefined term in both that makes it impossible to know what they mean until it is defined. I’m referring, of course to that undefined term “…”

    Or to be less snarky, leaving out the leftmost “sides” of these 6 (yes!) equations, since Cauchy we define the term involving that crazy lazy eight (one of the many meanings of infinity) in terms of a limit, which is to say in terms of a first-order statement involving functions, variables, constants, quantifiers and operators both boolean and arithmetic. Then we assume the axioms of the real number system and of classical (not intuitionistic) logic. Then I can prove that both of those two lines are true.

    Yet Euler didn’t need all that apparatus because he was reasoning formally and with an intuition unparalleled among mathematicians before or since, when he calculated this sum involving pi, solving the famous Basel problem. It wasn’t at all intuitive before Euler.

    It would still be nice to have an intuition about why reciprocal squares sum to a simple algebraic expression involving, mysteriously, pi. For example, I have an intuition about Buffon’s Needle approximating pi via probabilities, because angles are involved. Where are the angles in the Basel problem?

    One paper I happened on by surfing the net suggests that there are two ways to find the pi. First, relating this infinite series to a trigonometric infinite series (i.e. Fourier series). Second, relating this infinite series to calculating the area under a circle. Since Fourier series weren’t on the scene until half a century later, I’m guessing that Euler related the Basel problem to the area of a circle. He certainly used definite integrals, which “are” areas. I’m not an Euler scholar, so I don’t know what was on his mind.

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