A Visual Attempt at 1 + 2 + 3 + 4 + 5 + ... = -1/12
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Publicado 2024-08-04
1+2+3+4+5+... = -1/12
that is, that the sum of all positive integers is -1/12. One problem with this is that it isn't true using the techniques of real analysis, so visualizing it can be challenging (not to mention that visualizing negative numbers is challenging in and of itself). Another challenge is that people get pretty angry with the various methods used to produce this claim.
However, we are able to make sense of this sum using a series of visual arguments that connect three different infinite sums and choose one value to assign to them. Here I try to show visual representations for this argument. While this argument has its inconsistencies, there are more legitimate reasons out there to indicate these values make some sense. In each case, there are valid reasons to assign these values, though the ones here aren't always the best.
This sum has appeared many times on YouTube and has created a lot of controversy and excitement. I am not claiming this to be a set fact; instead, my intent is to show how one standard (though with problems) argument for this claim can be made visual in some sense. If you want to know more about the intricacies involved in this argument, see the videos linked below, especially the ones from Mathologer.
If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/visualproofs Thanks!
If you want to find other nice commentaries about this infinite sum, check out the following.
From @numberphile
• ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... =...
• Sum of Natural Numbers (second proof ...
• Why -1/12 is a gold nugget
• The Return of -1/12 - Numberphile
• Does -1/12 Protect Us From Infinity? ...
• One minus one plus one minus one - Nu...
From @Mathologer :
• Ramanujan: Making sense of 1+2+3+... ...
• Numberphile v. Math: the truth about ...
From @blackpenredpen
• Not -1/12
• the most famous Ramanujan sum 1+2+3+....
The alternating geometric series argument comes from a proof by The Viewpoints 2000 group from the October 2001 issue of Mathematics Magazine page 320 (www.jstor.org/stable/2691106 ).
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#manim #maths #mathematics #ramanujan #ramanujansummation #infinitesum #sumpositiveintegers #arithmeticmean #series #infiniteseries #divergentseries #divergent #convergent #geometricseries #abelsum #somepi
Todos los comentarios (21)
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"one way to use visual arguments to get at the proposed result" - I do appreciate the careful use of language here that doesn't overpromise that you've demonstrated a complete and airtight rigorous mathematical proof.
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This man can do visual proof of anything
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Before watching the video: There's no way he can convince me but let's see what he has to come up with After watching the video: YOU DIVERGENT SON OF A-
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Math is just counting an infinite amount zeros.
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5:32 Not sure if this is allowed. Visually, it seems to converge to 1. Graphically however, it's undefined, as both limit from left is different from limit on right. If you want to do that, you have to make sure that the diagonal shifting from the outer side also converge to that same point.
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The Analytic continuation of the Riemann zeta function gives the value of -1/12 for s = -1. This does not mean that the summation of all natural numbers is -1/12 - the whole point of analytic continuation is that you are extending the function to the domain where the original function is not defined.
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A=1/2 is completely wrong. It's a divergent series with infinitive numbers of+1 and -1. You can make it equal to anything by carefully grouping the terms. e.g. to make it equal to 3 group terms like this (1+1+1+1-1)+(1+1+1+1-1)+...
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Using the same logic at the end you can argue it equals whatever number you want. When you are dividing an infinite number into different groups, you are always going to end up with equal groups.
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In some sense, 1/2 begins to describe some kind of ratio regarding the number of negative and positive numbers in A. Whatever the definition of that ratio is, I can see the argument following from there. But that is not saying that the sum is -1/12. Rather some aspect of that sum can be numerically described.
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9:14 S - N = 4S I like how this implies that you have an infinite sum, you subtract a finite amount from it, and the sum gets bigger 😂😂
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I had no idea how you'd pull it off but this is beautifully and elegantly done! Incredible work!
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Divergent series are the new "imaginary" numbers. We must accept them and start learning how to work with them, what can or cant be done, instead of ignoring or avoiding it. Great video, congrats
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i've seen the algebraic version of this proof (the one that uses S, A, and N), but seeing it adapted into a visual form is a delight!
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so it's just the numberphile argument but with more squares
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Notice: Midwits are swarming the comment section. Yes, we all know the series doesn’t converge, and it is stated clearly in the video from the beginning
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6:42 there could be a problem with that. Order is gonna change the value of the infinite sum.
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I think this video actually shows that if the Grandi's series is convergent, the sum of all natural number also is. Some still get mad when we talk about those series convergence, but it is just a minor mathematical artifact. Chose your axioms and you can decide if the series diverges or converges. However the real question is not if the sum of natural numbers is convergent, but why -1/12 pops up in so many different ways. Do the mathematical operations that we perform have a underlying equivalence and we are always just getting the same "wrong" result? Or is there a -1/12 hidden behind the infinity? Great video nevertheless!
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As a programmer, the idea of adding up positive numbers and ending up with a negative result isn't that weird. It happens all the time, and we call it integer overflow.
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but isn't the limit not the real value? a limit is just a limit?
