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How to explain to a middle-school student the notion of a geometric series without any calculus (i.e. limits)? For example I want to convince my student that $$1 + \frac{1}{4} + \frac{1}{4^2} + \ldots + \frac{1}{4^n} = \frac{1 - (\frac{1}{4})^{n+1} }{ 1 - \frac{1}{4}}$$ at $n \to \infty$ gives 4/3?

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    $\begingroup$ @SujaanKunalan Depends on the 14 year old, and how much rigorous math they know. $\endgroup$ – Laertes Mar 11 '15 at 13:24
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    $\begingroup$ Don't do it; you'll ruin his/her remaining childhood. $\endgroup$ – Quinn Culver Mar 11 '15 at 14:18
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    $\begingroup$ You're certainly not going to explain "as $n\to\infty$ gives $4/3$" without any concept of limit -- since the thing you want to explain is a limit, you can't even say it without a (perhaps implicit) concept of limit. $\endgroup$ – Henning Makholm Mar 11 '15 at 15:50
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    $\begingroup$ @QuinnCulver or enhance it significantly :) $\endgroup$ – Jeel Shah Mar 11 '15 at 17:19
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    $\begingroup$ I have no idea whether this would be helpful or utterly baffling to students, but the two trains puzzle can be solved by a nice shortcut, understandable to most anyone - but it can also be solved by geometric series, and if you equate the latter with the former, you can figure out anything you want about geometric series. $\endgroup$ – Milo Brandt Mar 12 '15 at 1:36

13 Answers 13

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The equality is equivalent to

$$ \sum_{k=1}^{n}\frac{1}{4^k}=\frac{\frac{1}{4}-\frac{1}{4^{n+1}}}{1-\frac{1}{4}}$$

Now multiply both sides with $(1-\frac{1}{4})$ and everything will cancel out in the LHS except the first and the last term which are indeed $\frac{1}{4}$ and $-\frac{1}{4^{n+1}}$

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  • $\begingroup$ ;-) what I meant. $\endgroup$ – mathie314 Mar 11 '15 at 13:36
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    $\begingroup$ I don't really see how this answers the question. The question was how to explain to a 14 year old why this converges to $\frac43$, not why the equality is true. $\endgroup$ – Jason Mar 12 '15 at 7:49
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    $\begingroup$ 1) Addition and multiplication are things which children learn at this age, at least I learned it in school at this age. And as far as I know children will learn things earlier and earlier in school (What I mean: school is harder now in comparison to our school-time). 2) I don't see any problem to explain a child that $a^n$ will be smaller and smaller if $n$ grows bigger and bigger. To understand this only multiplication should be known which is no problem in this age. For example you could give him the numbers $a=0.5$ and $a=2$. Then let him calculate $a^n$ for different $n$. $\endgroup$ – Marm Mar 12 '15 at 8:02
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This could be explained using algebraic transformation but i would rather show a very simple geometric proof for sum:

1 + 1/2 + 1/4 + ... = 2

enter image description here

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    $\begingroup$ If we cut the remaining rectangle in $3/4$ at each step, we get the sum with common ratio $1/4$. The student could discover the relation by himself. This visualisation also works for proving the formula for the finite sum: Ask "What is the size of the remaining rectangle after $n$ steps?". $\endgroup$ – filipos Mar 11 '15 at 18:59
  • $\begingroup$ when i think of geometry, i think of pictures. this answer best IMO, especially when reinforced with Umberto P.'s answer $\endgroup$ – Evorlor Mar 11 '15 at 19:02
  • $\begingroup$ For a slightly more sophisticated student who'd like a proof that works for all ratios, I highly recommend the geometric proofs found in Nelsen's /Proofs Without Words/—my favorite is books.google.com/… . There's also a lovely proof at www41.homepage.villanova.edu/robert.styer/Bouncingball/… . I unfortunately can't find the source for it, although it does cite another source for the same proof. $\endgroup$ – Vectornaut Mar 12 '15 at 2:52
  • $\begingroup$ More proofs like that: deltami.edu.pl/temat/matematyka/geometria/planimetria/2011/06/… it is in Polish but I think that the images and formulas are straightforward to everyone. $\endgroup$ – Adam Baranowski Apr 24 '15 at 16:45
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I think a 14 year old can grasp the fact that $$\frac 12 + \frac 14 + \frac 18 + \frac 1{16} + \cdots = 1$$ rather intuitively. (Go halfway there, then half the remaining distance, then halfway again, and so on and you get arbitrarily close....)

If you are willing to do a little algebra (and wave hands about rearrangement) you get $$ 2 \left( \frac 14 + \frac 1{16} + \cdots \right) + \left( \frac 14 + \frac 1{16} + \cdots \right) = 1$$ so that $$ \frac 1{4} + \frac{1}{16} + \cdots = \frac 13. $$

Now add 1.

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    $\begingroup$ I can agree in principle, but I would not be so licentious with the rearrangement of an infinite sum. Also because they are 14 years old and may believe that these rearrangements are licit. $\endgroup$ – Oscar Mar 11 '15 at 13:28
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    $\begingroup$ @UmbertoP. Should the OP explain to the 14 years old that this is OK for positive series but not OK in general, and why does this happen? Or should the 14 years old figure that himself? The rearrangement of series, positive or not, is something subtle... $\endgroup$ – N. S. Mar 12 '15 at 6:38
  • $\begingroup$ @UmbertoP. +11 (and counting) for possibly misleading a 14 year old (who will likely not know see the importance of the distinction) that rearranging an infinite series is always valid. Oh boy. $\endgroup$ – Laertes Mar 12 '15 at 11:06
  • $\begingroup$ @UmbertoP. I'm afraid that I was referring to your answer. $\endgroup$ – Laertes Mar 12 '15 at 11:11
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Given a series $$S_n=1+x+x^2+\cdots+x^n$$, we have $$xS_n=S_{n+1}-1=S_n+x^{n+1}-1$$, so $$S_n(x-1)=x^{n+1}-1$$, or $$S_n={x^{n+1}-1\over x-1}={1-x^{n+1}\over 1-x}$$, which is the desired result.

For the infinite series, without using limits we see that if $$S=1+x+x^2+\cdots$$, then $$xS=S-1$$ (this is essentially the limit but is easy to see without formal calculus) and then $$S(x-1)=-1$$ and so $$S=\frac 1{1-x}.$$

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    $\begingroup$ Does not involve arbitrary rearrangement of infinite series so +1 $\endgroup$ – abligh Mar 14 '15 at 9:18
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$$ \frac{1}{3}-\frac{1}{4}=\frac{1}{12}=\frac{1}{4}\cdot\frac{1}{3} $$ so $$ \frac{1}{3}=\frac{1}{4}+\frac{1}{4}\cdot\frac{1}{3} $$ Now ask your 14 year old to plug in this expression for $1\over 3$ into itself, quite funny, bewildering and strange at first sight: $$ \frac{1}{3}=\frac{1}{4}+\frac{1}{4}\cdot \left(\frac{1}{4}+\frac{1}{4}\cdot\frac{1}{3}\right)= \frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^2}\frac{1}{3} $$ Repeat two or three times, then discuss the difference $$ \frac{1}{3}-\left(\frac{1}{4}+\frac{1}{4^2}+\dots+\frac{1}{4^n}\right) $$

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For example I want to convince my student that

this is impossible without talking about the notion a limit. How do you convince a student that $1,{1\over 2},{1 \over 3},\dots$ goes to zero? What does goes to even mean? In this situation $$ 1+\frac{1}{4}+\frac{1}{4^2}+\dots=\frac{4}{3} $$ You cannot convince somebody that this is true without defining the meaning of these little dots on the left side.

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What about multiplying the LHS by $(1 - \frac{1}{4})$?

Or is that what you wanted to avoid?

I mean it is not so difficult to understand that nearly all terms cancel... ?

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As I understand your question, your student know and understand the formula for a finite geometric sum.

You just need to convince them that the sum goes to $\frac 1{1-q}$ as $n \to \infty$.

Well the only thing left to do really is to convince them that if $x < 1$, then $x^n \to 0$.

You can do so by asking them to bring their calculator to the class, hit $1/2 \cdot 1/2$. Then again $1/2$. Then again $1/2$. After 10 or 20 iterations you can write this number out with decimal digits and they should grasp the fact that it's very close to $0$ (better to write the number out than to be left with something like $\frac 1{2^{16}}$ which may be less clear)

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  • $\begingroup$ Probably even easier to start with 1/10. It is 'obvious' that 0.1, 0.01, 0.001 ... "goes to 0" (or "gets as small as you like", if you prefer). $\endgroup$ – Alchymist Mar 12 '15 at 12:02
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We need to get rid of the idea that the average 14 year old is not yet not old enough to do abstract math. The undisputable fact is that the older you are the more difficult it becomes to learn it. If the average 14 year old would really struggle to understand the simple math needed to sum a geometric series, then how come they can operate their smartphones with ease?

So, I would say you could just do the summation of the first n terms using the standard algebraic method e.g. given in Leartses's answer. And then you argue that the limit is 4/3 by considering the difference between the finite sum of the first n terms. You show that for every $\epsilon>0$, no matter how small, there exists an N such that for all n > N the difference is smaller than $\epsilon$. There are many ways to explain this by drawing pictures. You can explain that if you replace 4/3 by another number then it this "game" of finding N for every $\epsilon$ will go wrong.

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  • $\begingroup$ Today I had no time to explain it to him in all details but I explained the idea which is in the hear of calculus, i.e. that we can add infinite contributions and still get the exact result. He could understand some logic behind it and he would seem able to tackle the idea that infinity is not a number, but surely these are concepts that take a lot of time to grasp. $\endgroup$ – Marion Mar 11 '15 at 20:43
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You happen to mention one of my favorite series. This may be convoluted, but consider those fraction in binary.

1/4 = .01 1/16 = .0001, etc. So your sum looks like .010101......

since S=.010101...

2S = .101010101....

and

3S = .111111.... which is 1, similar to .9999.... being 1.

If 3S=1, S=1/3

No calculus, and binary always comes in handy, in my opinion.

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Perhaps better to teach growth rather than decay. An old story goes...A king wanting to please his subject asked him what he wished for.The clever subject replied "Give me a checker-board, put 1 grain of wheat in the the first square, two in the second, four in the third to fill up the entire board (until $2^{64}$ grains in last square) to which the king readily agreed, until all go-downs ran out of wheat..

Calculation of compound interest can be also instructive.If money doubles every 5 years how much amount after 10 years and so on..

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  • $\begingroup$ This story is slightly improved by stating the subject invented chess, and the king's offer was his reward (since it demonstratess the inventor of chess was clever in more ways than one! ) $\endgroup$ – Kyle Hale Mar 11 '15 at 14:52
  • $\begingroup$ Yes, he put maths to proper use! $\endgroup$ – Narasimham Mar 11 '15 at 16:03
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Just explain that if the sum of the infinite series is $x$, then $4(x - 1) = x \implies x = 4/3$.

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If your $14$ year old is willing to accept without proof that the sum of a geometric series with positive ratio $\alpha<1$ exists at all (in your example for $\alpha=\frac14$), then I would argue as follows. Clearly all terms after the initial term$~t_0$ represent $\alpha$ times the total sum$~S$, since the terms taken in order have all been multiplied by$~\alpha$ with respect to the original terms. But then $t_0$ must be equal $1-\alpha$ times$~S$. But that makes $$S= \frac{t_0}{1-\alpha}.$$ In other words you don't need the finite sum to get at the infinite sum. One can however recover the finite sum as the infinite sum minus the part of the infinite sum excluded from the finite sum. If $t_{n+1}$ is the first term excluded, then this gives for the finite sum $$ \frac{t_0}{1-\alpha}-\frac{t_{n+1}}{1-\alpha}=\frac{t_0-t_{n+1}}{1-\alpha}=t_0\frac{1-\alpha^{n+1}}{1-\alpha}, $$ which is your sum. Actually like this it would be nicer and more natural to call the first term excluded $t_n$, and let the sum have $n$ rather than $n+1$ terms, but I've adapted to the notation of the question.

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While I like the graphical examples as a way of explaining this, I think the numerical intuition might be easier if you went for examples with nice decimal expansions.

For example, the series $9 + 0.9 + 0.09 + 0.009 +\dots $ can be seen to get closer and closer to $10$ (even if the precise, limiting sense in which it goes towards $10$ is harder for a 14 year old to understand).

If your child is familiar with the decimal expansion of one third (and if they don't, looking at the recurring decimal by manually performing the short division would be instructive) then $0.3 + 0.03 + 0.003 + 0.0003 + \dots$ also works well. Similarly for any fraction over nine, e.g. $\frac{7}{9}$ as the limit of $0.7 + 0.07 + 0.007 +\dots$

Such examples are, in my experience as an educator, a good way to introduce the concept of an infinite series whose partial sums approach a limit. That seems to be the major conceptual hurdle here. The actual algebra can come later, and not necessarily very much later. If they know the formula for the sum of the first $n$ terms of the series, then the limiting case can be established by considering the behaviour of $r^n$ as $n$ increases, when $-1 < r < 1$. I'd suggest that you deal with the effects of (repeated) multiplication by numbers between $0$ and $1$ as a prerequisite, before you broach the subject of geometric series. When exploring the limiting case algebraically, you can confirm it works on some examples with "obvious" answers, such as those listed above.

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