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Thank you John. My favorite Ramanujan’s formula: 1 + 2 + 3 + 4 + ⋯ = −1/12

by MrXOR 6y ago

RHSeeger 6y ago
That one is kind of silly.
- gus_massa 6y ago
  It is weird, very weird. A very nice explanation of the proof is in a video by Mathologer "Ramanujan: Making sense of 1+2+3+... = -1/12 and Co." https://www.youtube.com/watch?v=jcKRGpMiVTw https://www.youtube.com/watch?v=jcKRGpMiVTw [spoiler alert] You can extend finite summations to infinite summations. The first extensions are nice and you get intuitive results, and are the extensions studied in Calculus in the university. But these extensions are not enough for this sum. You must make more bold extensions, and the results are not as intuitive, and not useful outside some special applications.
  - RHSeeger 6y ago
    Honestly, I don't buy the proof. When you take a formula that's valid for a certain range of inputs and just use it outside that range anyways, you're no longer in the land of proof. At that point, you're in the land of "if we extend out mathematical system, we get this". And that's fine, but you don't wind up with "sum(1...inf) === -1/12", you wind up with "in this other mathematical system, sum....".
    - gus_massa 6y ago
      I'm not going to blame you. As the video explain, this is not a straightforward and obvious generalization of summation. I think that the generalization that use averages are fine, and some analytic continuations using power series are fine. Analytic functions are just too good to ignore them. It is not another mathematical system. It it just the extension of our mathematical system. Anyway, to get a result for this sum, you must drop too many of the obvious and expected properties of summation. For example if you add a zero in front of the sum, the result changes :(. There was a nice blog post I can't find, but there is a hint of the problem in https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF#Failure_of_stable_linear_summation_methods https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B... . So the result is ... polemic. Anyway, in some courses of a Math degree, one of the important ideas is that you have many possible definitions of convergence. For example, if you have a sequence of functions, you have https://en.wikipedia.org/wiki/Uniform_absolute-convergence https://en.wikipedia.org/wiki/Uniform_absolute-convergence and https://en.wikipedia.org/wiki/Weak_convergence_(Hilbert_space) https://en.wikipedia.org/wiki/Weak_convergence_(Hilbert_spac... and many many many more. So you don't write lim f_n -> f, you must write lim_{something} f_n -> f where something explain which definition of convenience you are using. The trick here is to hide the nasty problem of the multiple definitions of convergence in the dots. So the correct statement of the problem is 1+2+3+4+..._{with a weird convergence}=-1/12 but it doesn't look as nice as 1+2+3+4+...=-1/12
      - MrXOR 6y ago
        Nice answer.
- MrXOR 6y ago
  Niels Abel: "The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes."
beervirus 6y ago
Pretty sure that one predates Ramanunan...
- MrXOR 6y ago
  Yes, it has a long history. Like most math formulas, it back to Leonhard Euler. Ramanunan proved this series with his summation technique. [1] https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF#Ramanujan_summation https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B... [2] https://medium.com/cantors-paradise/the-ramanujan-summation-1-2-3-1-12-a8cc23dea793 https://medium.com/cantors-paradise/the-ramanujan-summation-...

Thank you John. My favorite Ramanujan’s formula: 1 + 2 + 3 + 4 + ⋯ = −1/12

by MrXOR 6y ago

Thank you John. My favorite Ramanujan’s formula: 1 + 2 + 3 + 4 + ⋯ = −1/12

RHSeeger 6y ago

That one is kind of silly.

gus_massa 6y ago
It is weird, very weird. A very nice explanation of the proof is in a video by Mathologer "Ramanujan: Making sense of 1+2+3+... = -1/12 and Co." https://www.youtube.com/watch?v=jcKRGpMiVTw https://www.youtube.com/watch?v=jcKRGpMiVTw [spoiler alert] You can extend finite summations to infinite summations. The first extensions are nice and you get intuitive results, and are the extensions studied in Calculus in the university. But these extensions are not enough for this sum. You must make more bold extensions, and the results are not as intuitive, and not useful outside some special applications.
- RHSeeger 6y ago
  Honestly, I don't buy the proof. When you take a formula that's valid for a certain range of inputs and just use it outside that range anyways, you're no longer in the land of proof. At that point, you're in the land of "if we extend out mathematical system, we get this". And that's fine, but you don't wind up with "sum(1...inf) === -1/12", you wind up with "in this other mathematical system, sum....".
  - gus_massa 6y ago
    I'm not going to blame you. As the video explain, this is not a straightforward and obvious generalization of summation. I think that the generalization that use averages are fine, and some analytic continuations using power series are fine. Analytic functions are just too good to ignore them. It is not another mathematical system. It it just the extension of our mathematical system. Anyway, to get a result for this sum, you must drop too many of the obvious and expected properties of summation. For example if you add a zero in front of the sum, the result changes :(. There was a nice blog post I can't find, but there is a hint of the problem in https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF#Failure_of_stable_linear_summation_methods https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B... . So the result is ... polemic. Anyway, in some courses of a Math degree, one of the important ideas is that you have many possible definitions of convergence. For example, if you have a sequence of functions, you have https://en.wikipedia.org/wiki/Uniform_absolute-convergence https://en.wikipedia.org/wiki/Uniform_absolute-convergence and https://en.wikipedia.org/wiki/Weak_convergence_(Hilbert_space) https://en.wikipedia.org/wiki/Weak_convergence_(Hilbert_spac... and many many many more. So you don't write lim f_n -> f, you must write lim_{something} f_n -> f where something explain which definition of convenience you are using. The trick here is to hide the nasty problem of the multiple definitions of convergence in the dots. So the correct statement of the problem is 1+2+3+4+..._{with a weird convergence}=-1/12 but it doesn't look as nice as 1+2+3+4+...=-1/12
    - MrXOR 6y ago
      Nice answer.
MrXOR 6y ago
Niels Abel: "The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes."

beervirus 6y ago

Pretty sure that one predates Ramanunan...

MrXOR 6y ago
Yes, it has a long history. Like most math formulas, it back to Leonhard Euler. Ramanunan proved this series with his summation technique. [1] https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF#Ramanujan_summation https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B... [2] https://medium.com/cantors-paradise/the-ramanujan-summation-1-2-3-1-12-a8cc23dea793 https://medium.com/cantors-paradise/the-ramanujan-summation-...