A visualization of why 1/4 + 1/16 + 1/64 + 1/256 + ... = 1/3 | Hacker News Reader

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

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200 pointsFlemlord47 comments

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.

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...

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.

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.

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).

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

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

I think the original example is more obvious.

There's also a merger of the two on Wikipedia:

http://en.wikipedia.org/wiki/File:Geometric_series_14_triang...

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?

No one color represents the series. They all do.

Yeah, so it would be equivalent to swap out any of the colors, which is why I stopped for a moment to think about why I automatically assumed black represented the series. It does make some sense visually, being the only of the three to occupy the diagonal, though of course mathematically, any of the three are equivalent. But the whole point was the visualization!

The white, black, and gray can all represent the series - they are equal in area! The triangular representation does convey the idea of "1/3ness" more naturally to me, but the white/gray/white scheme seems to throw off the comparison.

The equilateral triangle divided into four smaller such triangles, and the square divided into four smaller squares both have advantages as representations. Hmm. Do any other simple geometric shapes easily divide into self-similar shapes?

If the triangle would get the same colour-scheme, then it would be equally obvious. Of course it would be better if it was animated.

In the original, I can't see the one third at all. I originally saw the grey as being the items being summed, but after a second look I think the white may be this (not that it matters, the black could be it as well). It took me a while to realize that the point of the colors is to show that there are 3 of each size square.

Try looking at the different "layers" of squares. Start with the three big ones at top-left, bottom-left, and bottom-right. See how exactly one of the three is white (or black or gray, just pick one it doesn't matter). Now you see that one-third of this L-shaped piece is white, but what about the remaining quarter? Well, it's just the same. Look the the gray, black, and white square in the same configuration at the left & bottom of that quarter. Obviously one-third of that "L" is also white. Continue until you are convinced the one-third of every remaining part of the square is white.

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?

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

What are the 4 equal sections?

There are four equal (modulo rotation) parts: three share a corner with the original triangle and a fourth in the middle does not share a corner with the original triangle. The one in the middle is highlighted grey as 'the third', the two triangle parts sharing the lower left and lower right corner of the original triangle are ignored and the triangle sharing the top corner of the original triangle is segmented further

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.

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

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.

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 :-)

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.

Infinite GP: a/(1-r)

Ah, true.

http://en.wikipedia.org/wiki/Geometric_series#Formula

This wouldn't happen on Digg.

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

Rats!

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

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)

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.

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.)

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.

Considering when you created your account, and the subject of your post, "troll" seemed like the most likely explanation. Sorry about that. (It's sort of a typical troll to make a technically incorrect statement and then watch people get all agitated as they correct you. In fact, it's one of my favorite trolls, right behind making a sarcastic statement which you know will be taken literally by half of the audience, a la "A Modest Proposal".)

Anyways, assuming that your comment was in earnest, the arrow is typically used for functions (or sequences). E.g.,

  1/n -> 0 as n->infinity

You could write:

  1/4 + 1/16 + ... + 1/4^n -> 1/3

But you would write

  1/4 + 1/16 + ... = 1/3

You wouldn't write (or at least I've never seen it)

  1/4 + 1/16 + ... -> 1/3

It's not really a mathematical issue, just a definitional one: the left hand side is considered a real number, not a sequence of real numbers (or function :N->R, or whatever).

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.

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

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.

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

Xeno's Paradox

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.

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.

Also, just because a bunch of puny humans manage to convince each other that the world must be a certain way doesn't mean it actually is.

That depends on what your meaning of the word 'is' is.

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.

Actually, neither story proves either claim, which is apparent since there are rigorously defined and perfectly self consistent mathematical theories for each case. Philosophers just don't like them because they involve actual mathematical definitions, so they hide their heads in the sand and pretend they don't exist.

From that Wikipedia article, a quote from Russell: Georg Cantor invented a theory of continuity and a theory of infinity which did away with all the old paradoxes upon which philosophers had battened. ... Philosophers met the situation by not reading the authors concerned.

In other words, it's very easy to argue that infinite processes, continuous space, or motion in space don't make any sense if you can't be bothered to learn how they're rigorously defined in the mathematical theory you're arguing about.

And FWIW, philosophy may have been attempting to describe the nature of the universe, but I can't come up with a single example of an actual physical result that's come from the field. You may argue that math and physics sprung from philosophy, but realize that those two disciplines provide most of philosophy's harshest detractors these days.

And as far as retroactive claims that if people had listened to Zeno we might have stumbled upon special relativity earlier? (from the second link) Flat out bull poopy. The "inconsistencies" that special relativity resolves have nothing to do with classical mechanics at all, they have to do with E+M, and without a well tested and reliable E+M theory and the Michelson-Morley experiment to directly show us that the speed of light is constant regardless of motion we would have dismissed relativity theory as far too strange to be true (which, amusingly enough and in spite of massive evidence to the contrary, is a claim I've heard straight from the mouth of a tenured philosophy professor at an Ivy League school). An infinite speed of light would lead to a perfectly valid classical mechanics, albeit one that we could (now) prove is wrong, and there's nothing more logically consistent about either relativity or quantum mechanics that would have led us to either one without strenuous experimentation and Real Science.

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!"

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