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

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

#12

Post by zackattack »

Oompa wrote: post #3

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

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?

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

#13

Post by zackattack »

jakkals wrote: post #11

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 :-)

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.

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

#14

Post by Oompa »

zackattack wrote: post #12
Oompa wrote: post #3

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

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.

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

#15

Post by swapspace »

zackattack wrote: post #13
jakkals wrote: post #11

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 :-)

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)

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

#16

Post by zackattack »

swapspace wrote: post #15
zackattack wrote: post #13

Earlier quoted context omitted.

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

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

#18

Post by Xichekolas »

dustmop wrote: post #10

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.

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

#19

Post by jonsen »

zackattack wrote: post #13
jakkals wrote: post #11

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 :-)

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.

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

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

#20

Post by jimmybot »

rottencupcakes wrote: post #7
jimmybot wrote: post #6

Earlier quoted context omitted.

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 re…

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!
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