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Jack_Ian
Big Endian

Posted: Mon Nov 01, 2010 8:45 pm    Post subject: 1

 Quote: 1 = 1 + 0 + 0 + 0 +  1 = 1 + (1 + -1) + (1 + -1) +  1 = 1 + 1 + -1 + 1 + -1 + 1 + -1 +  1 = 1 + 1 + (-1 + 1) + (-1 + 1) +  1 = 1 + 1 + 0 + 0 + 0 +  1 = 2
I can see the step where the mistake was made, but I don't know why it was mathematically incorrect to take that step.
Can anyone shed any light on the rule that was broken?
wordcross

Posted: Mon Nov 01, 2010 8:59 pm    Post subject: 2

When you first add the pieces in parentheses, you have an infinite number of sets. When you remove them, then you are ignoring the fact that they are sets. It should really look like this:
 Quote: 1 = 1 + 0 + 0 + 0 +  1 = 1 + (1 + -1) + (1 + -1) +  1 = 1 + 1 + -1 + 1 + -1 + 1 + + -1 1 = 1 + 1 + (-1 + 1) + (-1 + 1) +  + -1 1 = 1 + 1 + 0 + 0 + 0 +  + -1 1 = 1

i.e. since they are sets, your series must always end in a -1
_________________
Has anyone really been far even as decided to use even go want to do look more like?
Jack_Ian
Big Endian

 Posted: Mon Nov 01, 2010 9:18 pm    Post subject: 3 In infinite set ending in a -1? I know the proof is wrong. My problem is that if I take any two consecutive lines, the step taken seems mathematically correct. It seems clear given the example above that the logic is incorrect, but how would I recognise such an error in a more complicated algebraic proof?
mith
Pitbull of Truth

 Posted: Mon Nov 01, 2010 9:41 pm    Post subject: 4 http://mathworld.wolfram.com/ConditionalConvergence.html
wordcross

Posted: Mon Nov 01, 2010 9:55 pm    Post subject: 5

 Quote: 1 = 1 + 0 + 0 + 0 +  1 = 1 + (1 + -1) + (1 + -1) + (1 + -1) +  1 = 1 + 1 + -1 + 1 + -1 + 1 + -1 +  1 = 1 + 1 + (-1 + 1) + (-1 + 1) + -1 +  1 = 1 + 1 + 0 + 0 + 0 + -1 +  1 = 1

Basically, when you add the step with the sets you're saying "Here's an infinite series, but after the first integer if you have a +1 then you must have a -1"
Then you proceed to step 3 which is fine, but then in step 4 you've ignored the "you must have a -1" part of what you said in step 2. So you have to show it.
_________________
Has anyone really been far even as decided to use even go want to do look more like?
Jack_Ian
Big Endian

Posted: Mon Nov 01, 2010 10:06 pm    Post subject: 6

 mith wrote: http://mathworld.wolfram.com/ConditionalConvergence.html
Thanks mith.

So does that make the very first step incorrect?
Is "(1 + -1) + (1 + -1) + (1 + -1) +  " not equal to zero after all?
Jack_Ian
Big Endian

Posted: Mon Nov 01, 2010 10:10 pm    Post subject: 7

 wordcross wrote: Basically, when you add the step with the sets you're saying "Here's an infinite series, but after the first integer if you have a +1 then you must have a -1" Then you proceed to step 3 which is fine, but then in step 4 you've ignored the "you must have a -1" part of what you said in step 2. So you have to show it.
So would you disagree with the following then?
 Quote: 1 = 1 + (1 + -1) + (1 + -1) + (1 + -1) +  1 = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) +
I'm basically looking for the general rule. mith has pointed me in the right direction.
wordcross

 Posted: Mon Nov 01, 2010 10:33 pm    Post subject: 8 Nope, that's fine. You still have a -1 for every +1 after the first._________________Has anyone really been far even as decided to use even go want to do look more like?
Nsof
Daedalian Member

Posted: Tue Nov 02, 2010 12:16 am    Post subject: 9

Jack_Ian wrote:
 Quote: 1 = 1 + 0 + 0 + 0 +  1 = 1 + (1 + -1) + (1 + -1) +  1 = 1 + 1 + -1 + 1 + -1 + 1 + -1 +  1 = 1 + 1 + (-1 + 1) + (-1 + 1) +  1 = 1 + 1 + 0 + 0 + 0 +  1 = 2
I can see the step where the mistake was made, but I don't know why it was mathematically incorrect to take that step.
Can anyone shed any light on the rule that was broken?
I think that this is where it got wrong
 Quote: 1 = 1 + (1 + -1) + (1 + -1) +
the serie on the right does not converge. Since it has no limit the equation is meaningless.
This also means that it is not conditional convergent. The Riemann series theorem (which states that a conditional convergent series can be arranged so that the series converges to any given value). does not apply.
On the other hand you can reach any integer value
_________________
Will sell this place for beer
MatthewV
Daedalian Member :_

 Posted: Tue Nov 02, 2010 12:34 am    Post subject: 10 This is why people usually will show the first few and the last few terms.
extro...*
Guest

Posted: Tue Nov 02, 2010 1:04 am    Post subject: 11

 Quote: 1 = 1 + (1 + -1) + (1 + -1) +  1 = 1 + 1 + -1 + 1 + -1 + 1 + -1 +

This step goes from an infinity of zeros, which adds up to zero, to two infinities, one of negative ones, another of positive ones, which can be paired up one-to-one, one-to-two, or many other ways. Thus it has the property that, in the words of the page mith linked to "by a suitable rearrangement of terms ... may be made to converge to any desired value, or to diverge"

 wordcross wrote: You still have a -1 for every +1 after the first.

And you also have a -1 for every two +1s, or three, etc., depending on how you want to group them. Infinities are like that. Thus again "by a suitable rearrangement of terms ... may be made to converge to any desired value, or to diverge".
extro...*
Guest

 Posted: Tue Nov 02, 2010 1:07 am    Post subject: 12
extro...*
Guest

Posted: Tue Nov 02, 2010 1:26 am    Post subject: 13

 extro...* wrote: Here: http://en.wikipedia.org/wiki/Grandi's_series

 Quote: It can be shown that it is not valid to perform many seemingly innocuous operations on a series, such as reordering individual terms, unless the series is absolutely convergent. Otherwise these operations can alter the result of summation. It's easy to see how terms of Grandi's series can be rearranged to arrive at any integer number, not only 0 or 1.
Jack_Ian
Big Endian

 Posted: Tue Nov 02, 2010 12:39 pm    Post subject: 14 Thank you extro That was exactly what I was looking for.
Jack_Ian
Big Endian

 Posted: Wed Nov 10, 2010 2:39 pm    Post subject: 15 I showed my 8 year old daughter the above proof and asked her where she thought the mistake was made. She told me that she wasn't sure about the third step, i.e. "1 = 1 + 1 + -1 + 1 + -1 + 1 + -1 +  ", and wanted to ask me a question first. Sure I said, ready to display my brilliance and become her mathematical hero. "Is infinity odd or even?" I was stumped. I said it was neither, but I could tell she wasn't impressed with that answer.
Chuck
Daedalian Member

 Posted: Wed Nov 10, 2010 2:48 pm    Post subject: 16 Let x = 1-1+1-1+1-1+1-1... subtract 1 from each side x-1 = -1+1-1+1-1+1-1+1... add the above two equations together 2x-1 = 0+0+0+0+0+0+0... = 0 add 1 to each side 2x = 1 divide both sides by 2 x = ½
ralphmerridew
Daedalian Member

 Posted: Wed Nov 10, 2010 4:54 pm    Post subject: 17 The short answer is that you have to justify that the sum of an infinite series is well-defined before you do anything related to that sum. Doing otherwise is akin to to "Let P be the point where lines L1 and L2 intersect. For reason X, P must be equidistant from points A and B ..." without first proving that lines L1 and L2 do intersect.
davidchatman
Icarian Member

 Posted: Tue Oct 25, 2011 9:57 am    Post subject: 18 Finally, here is the Nobel prize nomination for mathematics
cb*
Guest

Posted: Mon Mar 19, 2012 4:53 am    Post subject: 19

 Jack_Ian wrote: I showed my 8 year old daughter the above proof [...] "Is infinity odd or even?" I was stumped. I said it was neither, but I could tell she wasn't impressed with that answer.
Jack, your daughter is my hero. She found a great way to express simply one of the paradoxes of infinity.

Also, infinity is simultaneously "neither odd nor even" (if you argue that it's not really a number in the first place, so the concepts don't apply), and "both odd and even", considering that there's "some number" for which inf'ty = 2 * said number, and inf'ty = 1 + 2 * said number.
yevalent
Icarian Member

 Posted: Tue Sep 18, 2012 10:18 am    Post subject: 20 Here is the correct sequence ... 1=2 never possible ....... 1 = 1 + 0 + 0 + 0 +  1 = 1 + (1 + -1) + (1 + -1) +  1 = 1 + 1 + -1 + 1 + -1 + 1 + + -1 1 = 1 + 1 + (-1 + 1) + (-1 + 1) +  + -1 1 = 1 + 1 + 0 + 0 + 0 +  + -1 1 = 1 (Hence Proved) ....
extropalopakettle
No offense, but....

 Posted: Tue Sep 18, 2012 1:43 pm    Post subject: 21 1 = 1 + (1 + -1) + (1 + -1) +  1 = 1 + 1 + -1 + 1 + -1 + 1 + + -1 How can you just stick a "+ -1" at the end?
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