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Strauss (Guest)
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Jan1501, 07:18 PM (EST) 

"I need a proof"

Hello, I am looking for a proof that: 1/2 + 1/4 + 1/8 + 1/16 + 1/32 ..... and so on for infinity ..... =1 I have some friends who are very, VERY difficult to convince. Thank you. 

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Mike Fox (Guest)
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Jan2501, 10:46 PM (EST) 

2. "RE: I need a proof"
In response to message #0

I don't think you are going to find one. The old puzzle about moving the football half the distance to the goal line on each play, how many plays will it take to cross the goal line. You can't cross the goal by moving half the distance on each move. you get closer but never arrive. As it is with the series of fractions, you will get close to one but never arrive. 

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Bo Jacoby
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Sep0502, 06:38 AM (EST) 

4. "RE: I need a proof"
In response to message #0

>I am looking for a proof that: >1/2 1/4 1/8 1/16 1/32 ..... >and so on for infinity ..... =1 The unknown number is always called x : x=1/2 1/4 1/8 ... First put the first term outside parentheses: x=1/2 (1/4 1/8 1/16 ...) Next move the factor (1/2) outside the parenthesis: x=1/2 (1/2)(1/2 1/4 1/8 ...) Then observe that the contents of the second pair of parentheses exactly equals your original series: x=1/2 (1/2)x Now multiply both sides of this equation by 2: 2x=1 x Finally subtract x from both sides: x=1 There you are!>I have some friends who are very, VERY difficult to >convince. My friends are also hard to convince ! (see the other thread on this subject) 

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Bo Jacoby
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Sep0902, 06:10 AM (EST) 

5. "RE: I need a proof"
In response to message #4

Strangely all plussigns disappeared in the above message. I try again: >I am looking for a proof that: >1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ..... >and so on for infinity ..... =1 The unknown number is always called x : x=1/2 + 1/4 + 1/8 + ... First put the first term outside parentheses: x=1/2 + (1/4 + 1/8 + 1/16 + ...) Next move the factor (1/2) outside the parenthesis: x=1/2 + (1/2)(1/2 + 1/4 + 1/8 + ...) Then observe that the contents of the second pair of parentheses exactly equals your original series: x=1/2 + (1/2)x Now multiply both sides of this equation by 2: 2x=1 + x Finally subtract x from both sides: x=1 >There you are! > >>I have some friends who are very, VERY difficult to >>convince. >My friends are also hard to convince ! (see the other thread >on this subject)


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GoldShadow
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Oct1402, 06:45 AM (EST) 

6. "RE: I need a proof"
In response to message #5

I disagree. It would never exactly equal one. It would equal 1(1/infinity), meaning that it would be .99999999999..., so there's an infinitely small difference between that answer and one. An infinitely small difference is not the same thing as no difference, I think. 

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RicBrad
Member since Nov1601

Oct1402, 09:20 AM (EST) 

7. "RE: I need a proof"
In response to message #6

>I disagree. It would never exactly equal one. It would >equal 1(1/infinity),Infinity is a difficult concept  it would never exactly equal one after any finite period of time but, in the limit, at infinity, it would exactly equal 1. > meaning that it would be >.99999999999..., so there's an infinitely small difference >between that answer and one. An infinitely small difference >is not the same thing as no difference, I think. I think it is. Using the same argument as before,
let X = 0.9999... so 10X = 9.9999... so 9X = 10X  X = 9.0000... = 9
so X = 1 exactlyWhen you say 1/infinity, what that means is "the limit, as n tends to infinity, of 1 / n" which is, in fact, zero. Rich 

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GoldShadow
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Oct1402, 09:38 AM (EST) 

8. "RE: I need a proof"
In response to message #7

I guess if it comes to that it's somewhat hard to debate any further. I would say that 1/infinity > 0, and that infinity * 0 still = 0. But then, it gets pretty weird when you stop and think about that... you can't exactly imagine one pie or pizza, divided up among an infinite number of people. =P Of course, it is still essentially one, either way. 

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GoldShadow
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Oct1402, 12:18 PM (EST) 

9. "RE: I need a proof"
In response to message #8

Oh, BTW, I just wanted to point something out. By the logic you are using, anything divided by zero would be infinity. This is actually an idea I had assumed, but I changed my mind recently. That would also mean that a vertical line has a slope of infinity. (Right now I think that the only line with a slope of "infinity" would be 89.9999... (repeating) degrees, assuming 0 degrees would be set at a positive xaxis and 90 degrees at the positive yaxis). Then, dividing by zero would have to be 'more' than infinity, which is why it would be impossible. Hm..) Does that make any sense? I also think that you can show infinity by 1/.0000... (repeating overscore thing)...1, though I'm not sure that's legal. 

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Compmacgyver
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Dec0507, 11:02 PM (EST) 

10. "RE: I need a proof"
In response to message #9

Dividing by zero is always undefined. What you are discussing is the idea that in the limit process e.g. for any constant C the limit of C/n as n tends to 0 is infinity. 

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MPJ
guest

Dec0607, 07:41 PM (EST) 

12. "RE: I need a proof"
In response to message #11

Why not go empirical? Assign a value of "1" to some distance, in a straight line, you want to travel. You first travel half the original distance, then half the remainder, etc. So the distance you are traveling is 1/2, 1/4, 1/8, etc. Don't argue about whether you should be able to get there or not; fact is, you do, and as you look back at that moment, you will see that 1 = 1/2 + 1/4 + 1/8 .... It'seems to me that this argument would have appealed to a Hellenist, if not a Hellene. 

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Mark Huber
guest

Dec0807, 12:46 PM (EST) 

14. "RE: I need a proof"
In response to message #12

>Why not go empirical? Assign a value of "1" to some >distance, in a straight line, you want to travel. You first >travel half the original distance, then half the remainder, >etc. So the distance you are traveling is 1/2, 1/4, 1/8, >etc. Don't argue about whether you should be able to get >there or not; fact is, you do, and as you look back at that >moment, you will see that 1 = 1/2 + 1/4 + 1/8 .... It'seems >to me that this argument would have appealed to a Hellenist, >if not a Hellene. Here's the difficulty with that approach. Consider all the points on that straight line. They have length 0. But the union of all the points on the line is 1. So if the length of the line can always be broken up into the sum of its lengths, I can add 0 to itself an infinite number of times to get 1! Not good. This is the heart of Zeno's paradox: this argument works for 1/2 + 1/4 + ... = 1, but not for sum over all points of 0 = 1. You are implicitly using what is known as "countable additivity" in your argument, the proof of which relies on the Induction Axiom. While most mathematicians use this axiom with impunity, it can never be proven empirically. Mark Huber 

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MPJ
guest

Dec0907, 06:41 PM (EST) 

16. "RE: I need a proof"
In response to message #14

But why worry about points of length zero? By definition, there's no particle of matter smaller than the smallest particle we know about. Line up a bunch of those particles and you've got a line. But what does emptiness, meaning anything which has no length, contribute to the length of the line? You can squeeze all the zerolength points you want directly between two "smallest particles," yet the twodiameter distance across the two smallest particles won't change a bit. You can even set up a onetoone match between all the zerolength points between two "smallest particles," and all zerolength points in the whole line of "smallest particles." So "the union of all points on the line is 1," as you say, but the union is also 17.6 or any other length you want. But if you can prove that any length has the same number of zerolength points as any other length, then your argument can never be falsified by any data, and an argument that can't be falsified has no place in good old empirical science. My real point is that your argument is consistent with modern math theory & practice, but if you're dealing with someone to whom it's empirically obvious that "a line isn't made up of points of length zero, it's made up of intervals with dimensions greater than zero," then how can your audience consider your argument relevant? 

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Marcus Bizony
guest

Dec1307, 07:36 AM (EST) 

17. "RE: I need a proof"
In response to message #0

Hi the following argument convinces me and has the virtue of being easily visualised. Imagaine a glass tank with capacity 1 litre but initially empty. Imagine also an inexhaustible supply of porters (a supply of inexhaustible porters woudl do just as well). The first brings 1/2 litre of water and puts it in the tank; the next brings 1/4 litre and pours it in, the next brings 1/8 .....Will the tank overflow? Clealry not. Therefore whatever is the sum 1/2 + 1/4 + 1/8 + .... (call it S) we know that S is not greater than 2. Now make a mark on the side of the tank, anywhere you like but not actually at the top. Wil this mark eventualy be below the level of the water? Clearly so. If the mark represents a time when the volume of water in the tank is M, then we are saying that if M < 1, then S > M. Trying with S in place of M we find that S < 1 implies S > S, which is a contradiction. Therefore S is NOT less than 1. So we have simultaneously that S is not greater than 1 and S is not less than 1  leaving only one possibility! 

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