
A visualization of why 1/4 + 1/16 + 1/64 + 1/256 + ... = 1/3 - Flemlord
http://en.wikipedia.org/wiki/File:Geometric_series_14_square2.svg
======
michael_nielsen
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.

~~~
jonsen
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_nielsen
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.

------
Oompa
I like this one better. [http://web.mat.bham.ac.uk/pgweb/random/2009/04/proof-
without...](http://web.mat.bham.ac.uk/pgweb/random/2009/04/proof-without-
words/)

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

~~~
jimmybot
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?

~~~
baddox
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.

~~~
sp332
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.

------
jakkals
I like this!

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

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

'Proved' by looking at a picture :-)

~~~
zackattack
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
Infinite GP: a/(1-r)

~~~
zackattack
Ah, true.

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

~~~
I_got_fifty
This wouldn't happen on Digg.

------
tenzero
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.

~~~
jibiki
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.)

~~~
tenzero
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.

D.

~~~
jibiki
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).

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

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

------
gabeybaby
Xeno's Paradox

~~~
LogicHoleFlaw
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
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.

~~~
321abc
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.

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

------
pbhjpbhj
hacker news needs to allow mathml!

