A visualization of why 1/4 + 1/16 + 1/64 + 1/256 + ... = 1/3

http://en.wikipedia.org/wiki/File:Geometric_series_14_square2.svg

by Flemlord • 17 years ago

200 47 17 years ago

en.wikipedia.org

Oompa2• 17 years ago

I like this one better. http://web.mat.bham.ac.uk/pgweb/random/2009/04/proof-without...

ramidarigaz• 17 years ago

I'm not so sure. It seems less immediately obvious.

malkia• 17 years ago

I think the original example is more obvious.

jimmybot2• 17 years ago

In the original, the (1/4)^n is more obvious to me, while in the second, the 1/3 part of it is more obvious.

But the coloring scheme in the two are different too. The first uses three colors, the second two colors. What if the light gray in the first was white instead? I think then the 1/3 might pop out better.

Wait a second, does everyone even see the same thing? Although it doesn't matter which color you pick to represent the sum of the geometric series, I defaulted to the black squares representing the series. In the second, I assumed the gray represented the squares of the series. How about others?

zackattack4• 17 years ago

This one seems immediately obvious of the thirdness, as other posters have pointed out. But I don't understand how each row ALSO represents 1/4^n?

Oompa2• 17 years ago

Since the triangle is broken up into 4 equal sections and one is highlighted.

pkulak• 17 years ago

I'm not sure how you show that the second triangle is half the volume of the first with that picture. With the Wikipedia proof, you really needed only the picture.

rottencupcakes2• 17 years ago

I like these trapezoids: http://math.rice.edu/~lanius/Lessons/Series/infinite.htm

lallysingh• 17 years ago

Indeed, the reason for the exact relation sigma (1/4^n) == 1/3 becomes more obvious. Split an area into fourths, use up one fourth, and another fourth of that fourth, and the whole set becomes an infinite series of triples, with the middle (shaded) one filled in.

dustmop• 17 years ago

If the grey were made white instead I would have gotten this immediately.

Xichekolas• 17 years ago

The point of having three colors is to show that there are three boxes of each size. As such, if you pick any of the colors to represent the series, it's plain to see that all the boxes of that color will make up 1/3 of the total area.

TheSOB88• 17 years ago

Too bad, that doesn't change the makeup of the average person's mind.

jakkals2• 17 years ago

I like this!

Immediately I can also see that 1/5 + 1/25 + 1/125 + ... = 1/4

To generalize: 1/x + 1/(xx) + 1/(xx*x) + ... = 1(x+1)

'Proved' by looking at a picture :-)

zackattack4• 17 years ago

You mean, = 1/(x-1) ;)

Can someone please post a non-visual proof of why this is the case? In the meantime, I am working on figuring out my own.

swapspace• 17 years ago

Infinite GP: a/(1-r)

jonsen2• 17 years ago

You can get the proof idea from the picture:

  1/4 is 1/3 of 3/4
  1/4 of 1/4 is 1/3 of 3/4 of 1/4
  etc.

In math:

  1/4   = 1/3*3/4
  1/4^2 = 1/3*3/4*1/4
  1/4^3 = 1/3*3/4*1/4^2
  etc.

Summing equations:

  (1/4^1 + 1/4^2 + ...) = 1/3 * 3/4 * (1 + 1/4^1 + 1/4^2 + ...)
  <=>
  (1/4^1 + 1/4^2 + ...) = 1/3 * 3/4 + 1/3 * 3/4 * (1/4^1 + 1/4^2 + ...)
  <=>
  x = 1/3 * 3/4 + 1/3 * 3/4 * x
  <=>
  x - 1/4 x = 1/4
  <=>
  3/4 x = 1/4
  <=>
  x = 4/3 * 1/4
  <=>
  x = 1/3

jakkals2• 17 years ago

Rats!

Yes, I did indeed mean 1/(x-1). Thanks for the correction.

ewjordan2• 17 years ago

Not a rigorous proof (I'll leave that to further investigation):

S(n) = 1/n + 1/n^2 + ...

= 1/n ( 1 + 1/n + 1/n^2 + ...) <--needs more justification in a rigorous proof

S(n) = 1/n ( 1 + S(n) )

Simple algebra from here:

n * S(n) - S(n) = 1

S(n) = 1 / (n-1)

gabeybaby• 17 years ago

Xeno's Paradox

LogicHoleFlaw2• 17 years ago

Gah, Zeno's Paradox. A professor tried to stump the class with that one in an introductory philosophy course I took. I then proceeded to introduce him the fundamental principles of calculus with respect to limits. I think I threw in some snark about how this was the difference between mathematicians and philosophers - mathematicians actually find solutions! Then scientists ensure they apply to reality, and engineers make them useful.

jordyhoyt• 17 years ago

Then, I'm afraid, you're missing the point of the paradox. There are a few stories (improperly called "paradoxes") attributed to Zeno: http://en.wikipedia.org/wiki/Zeno%27s_paradoxes

You're probably thinking of The Dichotomy. This story points out that matter must not be infinitely divisible. The paired story, The Arrow, shows that a universe of finite, indivisible pieces is also impossible. Thus, Zeno's paradox.

Further, very interesting reading: http://www.mathpages.com/rr/s3-07/3-07.htm

Math just gave us a way of coming up with the obvious answer, it does not describe the nature of the universe, which is what the philosophy was attempting.

dfranke• 17 years ago

The problem with philosophers is not that they can't find solutions, it's that they're incapable of rejecting bad ones. There are plenty of good ideas in philosophy, but you can't pick them out from the flood stupid ones without thinking everything through from square one on your own. Hence the reason that millenia later we're still here debating Zeno, while physicists no longer have to bother giving much thought to 100-year-old rejected theories.

A university dean approaches the chair of the Physics department and tells him, "Look, we really need to talk your budget. Every year it's particle accelerator this and supercomputer that. You're bleeding the college dry. Why can't you be more like the Math department? All they ever ask us for is pencils and chalkboards and wastebaskets. Or better yet, why can't you be like the Philosophy department? They don't even ask for wastebaskets!"

michael_nielsen2• 17 years ago

A very cute algebraic way to see this is to do arithmetic in base 4. In base 4 the series on the left is:

0.1 + 0.01 + 0.001 + ... = 0.111... (recurring)

Multiplying the right hand side by 3 gives 0.3333... = 1, and so the original series must have just been 1/3.

jonsen2• 17 years ago

Very nice indeed, but you can do it in binary, where the series is

  0.01 + 0.0001 + 0.000001 + ... = 0.010101010101...

Multiplying the right hand side by 2 gives

  0.101010101010...

    0.010101010101...
  + 0.101010101010...
  -------------------
    0.111111111111... = 1 = 11*0.010101010101...

michael_nielsen2• 17 years ago

The advantage of my proof is that it generalizes to give the sum of 1/n + 1/n.n + ... Just replace 4 everywhere by n, and 3 by n-1:

1/n + 1/n.n + ... = 0.1111... (in base n)

Multiply by (n-1) to get:

0.(n-1)(n-1)(n-1)... = 1

So the original sum must have been 1/(n-1).

This is not so easy in binary.

ftse• 17 years ago

Cute. To illustrate the sum of limits, however, I prefer a graphic with sum -> 1. Take an empty circle then add a chunk of shaded semicircle, then add one quarter, then add one eighth... you get the idea.

eru• 17 years ago

With this method you won't be able to tell converging sequences from diverging ones. Suppose I used your circle and put in the first hundred terms of the harmonic sequence (divided by 10 or so, to fit in the unit area).

tenzero2• 17 years ago

While pedantic, the above "expression" is NOT = 1/3

1/3 is the limit, as the sum of n=1 to n -> infinity, of (1/4)^n

The "result" converges towards 1/3. You can get as close to 1/3 as you like, but the result will never quite equal 1/3.

Cheers Dion.

jibiki2• 17 years ago

I am on the fence about whether this is a good troll or a bad troll. It certainly exploits the "someone on the internet is WRONG" ethos of HN, but I don't think it does so in a particularly amusing way. I'm going to say that it's a rather boring troll.

(I do find myself compelled to say that the mathematical convention is that an infinite sum is defined to be equal to the limit of the partial sums, if it exists.)

tenzero2• 17 years ago

Well perhaps I delivered the statement poorly. My intent is not to troll, I detest Trolls.

The convention I have seen is to use the symbol of an arrow such as -> to denote the concept of approaching.

However I did state I was being pedantic.

yuushi• 17 years ago

Yes, it is 1/3. Given a series of partial sums (which in turn form an infinite sequence), then if the infinite sequence converges to some number B, the infinite sum likewise converges to the same number B.

It's exactly like saying lim as x -> 1 of 2x -> 2. You don't write it that way. You write it as lim as x -> 1 of 2x = 2.

pbhjpbhj• 17 years ago

hacker news needs to allow mathml!

A visualization of why 1/4 + 1/16 + 1/64 + 1/256 + ... = 1/3

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