Is 1+2+3+4+... equal to -1/12? A mind-boggling proof!

  • Thread starter fargoth
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In summary, the conversation is about a proof the speaker saw that appears to contradict their understanding of series convergence. They mention a rule about comparing the sums of two series and question its validity. They also bring up the concept of series divergence and ask about the implications of a_n not equaling zero as n approaches infinity. They express confusion about how the proof in question can result in a negative sum. The speaker then provides a counterargument and asks for clarification on where the other person's reasoning may be incorrect.
  • #1
fargoth
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I've see this neat proof:
http://www.youtube.com/watch?v=E-d9...eature=iv&annotation_id=annotation_3085392237 (for some reason the youtube tag didn't work in preview...)
And now I don't see how what I've learned about series convergence is true...
I've been told that if [itex]a_n > b_n \forall n[/itex] then [itex] \sum a_n > \sum b_n [/itex] therefore, if [itex] \sum b_n [/itex] is divergent then, [itex] \sum a_n [/itex] must be too.
Also, If the partial sum diverges, the series is said to be divergent, isn't it?
And what about [itex] a_n \neq 0 [/itex] for n that tends to infinity?
So many ways I could show this series diverges, yet he show it's equal to -1/12?

Where am I, or is he, wrong?
 
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  • #2
I think this cannot be true. The sum of all natural numbers up to N equals (as also shown in the end of the video) ## N(N+1)/2 ##. This obviously goes to infinity as N goes to infinity. And of course there is also no way how strictly positive numbers can add up to give a negative result.
 
  • #3
That's what I was saying :)
So where is he wrong?
 
  • #4
  • #5
And as this link was posted we can safely close the thread.
 

1. What is the meaning of "1+2+3+4+... = -1/12"?

The equation "1+2+3+4+... = -1/12" represents the sum of all positive integers (1, 2, 3, 4, etc.) to infinity, which is equal to -1/12. This result is counterintuitive and requires a deeper understanding of mathematical concepts.

2. How is the equation "1+2+3+4+... = -1/12" proven?

The proof of this equation lies in the mathematical concept of summation and the Riemann zeta function. By manipulating the zeta function and using complex analysis, mathematicians have shown that the infinite sum of positive integers can indeed equal -1/12.

3. Why does the equation "1+2+3+4+... = -1/12" seem to contradict basic arithmetic?

At first glance, the equation does seem to contradict basic arithmetic, as we are taught that summing positive integers should always result in a positive number. However, this equation deals with the concept of infinity and requires a more advanced understanding of mathematics to fully grasp.

4. What practical applications does the equation "1+2+3+4+... = -1/12" have?

The equation has several practical applications in fields such as physics, string theory, and number theory. It is used to solve problems related to infinity, and its results have been used to make predictions in quantum mechanics and other scientific theories.

5. Is it accurate to say that "1+2+3+4+... = -1/12"?

The equation "1+2+3+4+... = -1/12" is accurate in the context of mathematical concepts and theories. However, it is important to note that this equation is not a traditional mathematical sum and should not be used in everyday calculations or applications.

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