• Astounding: 1+2+3+4+5+... = -1/12
    123 replies, posted
[QUOTE=Laserbeams;43499202]∞-∞ does not equal 0. These paradoxes are really fucking with my mind[/QUOTE] It should if in this case the infinities are equal to each other, I'm not doing infinity + 1 = infinity - infinity = 0 kind of things. And if you follow the 1-1 2-2 3-3... pattern it would always get 0, shifting it makes that visualization impossible, but shifting doesn't change the number so it would be 0.
The only problem I see is that in the first sum they just kinda take 1/2 and roll with it, while I have a tendency to say that it is either 1 or 0, depending on where you would get a theoretical cut-off point that never comes. (Or hell, even both as some sort of Schrodinger's cat style of thing.)
While we're on the topics of misinformation and ambiguity, try find the problem in this! [IMG]http://i.imgur.com/YBhcUuv.png[/IMG]
[QUOTE=Quiet;43499718]While we're on the topics of misinformation and ambiguity, try find the problem in this! [IMG]http://i.imgur.com/YBhcUuv.png[/IMG][/QUOTE] (e^2*pi*i)^i != e^2*pi*i*i [editline]11th January 2014[/editline] [img]http://latex.codecogs.com/png.latex?%5Ctext%7Blet%20%7D%20S%28n%29%20%3D%20%5Csum_%7Bi%3D1%7D%5E%7Bn%7Di%2C%5C%3A%20n%20%5Cgeq%200%5C%5C%20%5C%5C%20%5C%5C%20S%28n+1%29%20%3D%20%5Cleft%28%5Csum_%7Bi%3D1%7D%5E%7Bn%7Di%5Cright%29%20+%20n%20+%201%5C%5C%20%5C%5C%20%5C%5C%20S%28n+1%29%20%3D%20S%28n%29%20+%20n%20+%201%20%5C%5C%20%5C%5C%20n%20%5Cgeq%200%20%5C%5C%20%5CRightarrow%20n+1%20%3E%200%20%5C%5C%20%5CRightarrow%20S%28n%29%20%3C%20S%28n%29+%20n%20+%201%20%5C%5C%20%5CRightarrow%20S%28n%29%20%3C%20S%28n+1%29%20%5C%5C%20%5CRightarrow%20S%28n%29%20%3C%20S%28n+k%29%2C%5C%3A%20k%20%3E%200%5C%3A%20%5Ctext%7B%20%28through%20induction%29%7D%5C%5C%20%5CRightarrow%20S%28a%29%20%3C%20S%28b%29%20%5Ciff%200%20%5Cleq%20a%20%3C%20b%20%5C%5C%20%5C%5C%20%5Ctext%7Bassume%20%7D%20S%28%5Cinfty%29%20%3D%20-%5Cfrac%7B1%7D%7B12%7D%20%5C%5C%20%5C%5C%200%20%5Cleq%200%20%3C%20%5Cinfty%20%5C%5C%20%5CRightarrow%20S%280%29%20%3C%20S%28%5Cinfty%29%20%5C%5C%20%5CRightarrow%20S%280%29%20%5Ctimes%20-12%20%3E%20S%28%5Cinfty%29%20%5Ctimes%20-12%20%5C%5C%20%5CRightarrow%200%20%5Ctimes%20-12%20%3E%20-%5Cfrac%7B1%7D%7B12%7D%20%5Ctimes%20-12%20%5C%5C%20%5CRightarrow%200%20%3E%201%20%5C%5C%20%5C%5C%200%20%5Cleq%201%2C%5C%3A%200%20%3E%201%20%5C%5C%20%5CRightarrow%200%20%5Cleq%201%20%3C%200%20%5C%5C%20%5CRightarrow%200%20%3D%201[/img]
[QUOTE=blacksam;43497954]Please, I get all the ladies doing math. "Would you like to see the exponential growth of my natural log?"[/QUOTE] eughhh cringeeee
[QUOTE=Ziks;43499949](e^2*pi*i)^i != e^2*pi*i*i[/QUOTE] True. [QUOTE=Ziks;43499949] [img]http://latex.codecogs.com/png.latex?%5Ctext%7Blet%20%7D%20S%28n%29%20%3D%20%5Csum_%7Bi%3D1%7D%5E%7Bn%7Di%2C%5C%3A%20n%20%5Cgeq%200%5C%5C%20%5C%5C%20%5C%5C%20S%28n+1%29%20%3D%20%5Cleft%28%5Csum_%7Bi%3D1%7D%5E%7Bn%7Di%5Cright%29%20+%20n%20+%201%5C%5C%20%5C%5C%20%5C%5C%20S%28n+1%29%20%3D%20S%28n%29%20+%20n%20+%201%20%5C%5C%20%5C%5C%20n%20%5Cgeq%200%20%5C%5C%20%5CRightarrow%20n+1%20%3E%200%20%5C%5C%20%5CRightarrow%20S%28n%29%20%3C%20S%28n%29+%20n%20+%201%20%5C%5C%20%5CRightarrow%20S%28n%29%20%3C%20S%28n+1%29%20%5C%5C%20%5CRightarrow%20S%28n%29%20%3C%20S%28n+k%29%2C%5C%3A%20k%20%3E%200%5C%3A%20%5Ctext%7B%20%28through%20induction%29%7D%5C%5C%20%5CRightarrow%20S%28a%29%20%3C%20S%28b%29%20%5Ciff%200%20%5Cleq%20a%20%3C%20b%20%5C%5C%20%5C%5C%20%5Ctext%7Bassume%20%7D%20S%28%5Cinfty%29%20%3D%20-%5Cfrac%7B1%7D%7B12%7D%20%5C%5C%20%5C%5C%200%20%5Cleq%200%20%3C%20%5Cinfty%20%5C%5C%20%5CRightarrow%20S%280%29%20%3C%20S%28%5Cinfty%29%20%5C%5C%20%5CRightarrow%20S%280%29%20%5Ctimes%20-12%20%3E%20S%28%5Cinfty%29%20%5Ctimes%20-12%20%5C%5C%20%5CRightarrow%200%20%5Ctimes%20-12%20%3E%20-%5Cfrac%7B1%7D%7B12%7D%20%5Ctimes%20-12%20%5C%5C%20%5CRightarrow%200%20%3E%201%20%5C%5C%20%5C%5C%200%20%5Cleq%201%2C%5C%3A%200%20%3E%201%20%5C%5C%20%5CRightarrow%200%20%5Cleq%201%20%3C%200%20%5C%5C%20%5CRightarrow%200%20%3D%201[/img][/QUOTE] The assumption is wrong.
[QUOTE=Quiet;43499718]While we're on the topics of misinformation and ambiguity, try find the problem in this! [IMG]http://i.imgur.com/YBhcUuv.png[/IMG][/QUOTE] You are ignoring the multi-valued phase. [IMG]http://quicklatex.com/cache3/ql_a920e42b7fdfa2aa511cc01c383aa294_l3.png[/IMG] Everything is fucked from that point, unless you choose a branch.
The funniest thing in all of this is: if, at some stop, we stop calculating the sum, it would be just equal to the sum of every number. So the trick works only for ζ(-1). I think the bottom line is: the result isn't absurd (it has actually been proven thorugh the very same zeta-function if you'd bother to watch the addendum video. Besides, the mentioned fact that it has real physical implications is pretty convincing), but VERY VERY non-intuitive. Mind-bendingly non-intuitive.
[QUOTE=Quiet;43500021]The assumption is wrong.[/QUOTE] Exactly, so S(infinity) != -1/12
Can someone explain how it's a negative value? (and how does the result as the camerman suggested, not tend toward infinity?)
[QUOTE=Ziks;43500063]Exactly, so S(infinity) != -1/12[/QUOTE] You're treating infinity as if it were a number. It's not.
[QUOTE=KillerJaguar;43500189]You're treating infinity as if it were a number. It's not.[/QUOTE] All I say is that infinity is greater than 0. S(infinity) = -1/12 is what the video is claiming.
[QUOTE=Ziks;43500206]All I say is that infinity is greater than 0. S(infinity) = -1/12 is what the video is claiming.[/QUOTE] Again, you're treating infinity as if it were a number. It's not. Infinity is neither greater than zero nor less than zero nor zero because it's not a real number.
[QUOTE=KillerJaguar;43500271]Again, you're treating infinity as if it were a number. It's not. Infinity is neither greater than zero nor less than zero because it's not a real number.[/QUOTE] Not true, infinity is defined as being greater than any real number. Infinity may not be a number, but it's a concept pretty much defined by comparisons to real numbers. [editline]11th January 2014[/editline] [IMG]http://puu.sh/6hr5H.png[/IMG]
[QUOTE=Ziks;43500299]Not true, infinity is defined as being greater than any real number.[/QUOTE] Whether or not you decide to use infinity as a number, the limit of S(n) as n approaches infinity is easy to prove to diverge. [IMG]http://i.imgur.com/IIFwGIR.png[/IMG]
[QUOTE=Ziks;43500299]Not true, infinity is defined as being greater than any real number. Infinity may not be a number, but it's a concept pretty much defined by comparisons to real numbers. [editline]11th January 2014[/editline] [IMG]http://puu.sh/6hr5H.png[/IMG][/QUOTE] That is true because: [quote]Infinity, most often denoted as [img]http://mathworld.wolfram.com/images/equations/Infinity/Inline1.gif[/img], is an unbounded quantity that is [b]greater than every real number.[/b][/quote] [url=http://mathworld.wolfram.com/Infinity.html]Source[/url]
[QUOTE=Quiet;43500324]Whether or not you decide to use infinity as a number...[/QUOTE] I never used infinity as a number. [editline]11th January 2014[/editline] [QUOTE=KillerJaguar;43500372]That is true because: [url=http://mathworld.wolfram.com/Infinity.html]Source[/url][/QUOTE] Surely that's precisely how I used it? What have I said that is incorrect? I stated the only comparison I used with infinity was that it was greater than 0, which at first you said I couldn't do but now you agree it is true? [editline]11th January 2014[/editline] [QUOTE=KillerJaguar;43500271]Infinity is neither greater than zero...[/QUOTE] [QUOTE=Ziks;43500299][IMG]http://puu.sh/6hr5H.png[/IMG][/QUOTE] [QUOTE=KillerJaguar;43500372]That is true...[/QUOTE]
[QUOTE=Ziks;43500374]Surely that's precisely how I used it? What have I said that is incorrect? I stated the only comparison I used with infinity was that it was greater than 0, which at first you said I couldn't do but now you agree it is true?[/QUOTE] Because you said infinity is greater than zero, which is incorrect. Infinity is greater than [b]any[/b] real number. To provide a counter example, if infinity is greater than zero, why is this false? [img]https://dl.dropboxusercontent.com/u/3877425/temp/infinity.png[/img]
[QUOTE=KillerJaguar;43500420]Because you said infinity is greater than zero, which is incorrect. Infinity is greater than [b]any[/b] real number. To provide a counter example, if infinity is greater than zero, why is this false? [img]https://dl.dropboxusercontent.com/u/3877425/temp/infinity.png[/img][/QUOTE] 0 is a real number. If ∀ x ∈ ℝ : ∞ > x, then surely ∞ > 0?
[QUOTE=Quiet;43500433]0 is a real number. If ∀ x ∈ ℝ : ∞ > x, then surely ∞ > 0?[/QUOTE] Again, you're treating infinity as if it were a real number.
[QUOTE=KillerJaguar;43500420]Because you said infinity is greater than zero, which is incorrect. Infinity is greater than [b]any[/b] real number. To provide a counter example, if infinity is greater than zero, why is this false? [img]https://dl.dropboxusercontent.com/u/3877425/temp/infinity.png[/img][/QUOTE] You might want to re-read your post. Surely the fact that infinity is greater than any real number supports the fact that it is greater than zero? Zero, after all, is a number, so is included within the set of any number.
[QUOTE=KillerJaguar;43500468]Again, you're treating infinity as if it were a real number.[/QUOTE] Claiming that infinity is greater than any real number is not?
[QUOTE=KillerJaguar;43500468]Again, you're treating infinity as if it were a real number.[/QUOTE] No, we are not. We are treating infinity as an abstract concept for which comparison with real numbers are permitted.
[QUOTE=Ziks;43500490]No, we are not. We are treating infinity as an abstract concept for which comparison with real numbers are permitted.[/QUOTE] To be fair, you did use S(∞). What did you mean by this? The limit of S(n) as n approaches infinity? I don't see any other definition.
[QUOTE=KillerJaguar;43500420]To provide a counter example, if infinity is greater than zero, why is this false? [img]https://dl.dropboxusercontent.com/u/3877425/temp/infinity.png[/img][/QUOTE] Why on earth would infinity being greater than zero imply this?
[QUOTE=Ziks;43500473]You might want to re-read your post. Surely the fact that infinity is greater than any real number supports the fact that it is greater than zero? Zero, after all, is a number, so is included within the set of any number.[/QUOTE] But then why is infinity less than 10 false? 10 > 0 infinity > 0 therefore infinity > 10 But then why does it say 10 > infinity > 0 is false
[QUOTE=Quiet;43500515]To be fair, you did use S(∞). What did you mean by this? The limit of S(n) as n approaches infinity? I don't see any other definition.[/QUOTE] Using infinity as the input for the function S(n), exactly as used in the video. I'm essentially claiming that you can't do that, S(∞) is undefined and certainly not -1/12. [editline]11th January 2014[/editline] [QUOTE=KillerJaguar;43500519]But then why is infinity less than 10 false? 10 > 0 infinity > 0 therefore infinity > 10 But then why does it say 10 > infinity > 0 is false[/QUOTE] Again, re-read what you just typed. Infinity is greater than all real numbers. 0 is a real number. 10 is a real number. So infinity > 0, and infinity > 10.
[QUOTE=KillerJaguar;43500519]But then why is infinity less than 10 false? infinity > 10 But then why does it say 10 > infinity > 0 is false[/QUOTE] I fail to see a contradiction. [editline]11th January 2014[/editline] [QUOTE=Ziks;43500526]Using infinity as the input for the function S(n), exactly as used in the video. I'm essentially claiming that you can't do that, S(∞) is undefined and certainly not -1/12. [/QUOTE] Unless the domain of the function includes some defined ∞ entity, you can't use it as an input.
[QUOTE=Quiet;43500540]Unless the domain of the function includes some defined ∞ entity, you can't use it as an input.[/QUOTE] Yep, so basically you would have to make a special definition for S that includes infinity in the domain. But for S as it stands there is no defined answer when giving infinity as input, which contradicts what the video states. If you do claim a real result when using infinity as input you will always get a contradiction, so that result is incorrect.
[QUOTE=Lazore;43496026]what i dont get, is how they can just move the second part of S2, what kind of rule is that?[/QUOTE] don't you remember the order of operations?
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