A Visual Attempt at 1 + 2 + 3 + 4 + 5 + ... = -1/12

I often get requests to give a visual proof of the claim that
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 want to find other nice commentaries about this infinite sum, check out the following.

From @numberphile

From @Mathologer :

From @blackpenredpen

To learn more about animating with manim, check out:

#manim #maths #mathematics #ramanujan #ramanujansummation #infinitesum #sumpositiveintegers #arithmeticmean #series #infiniteseries #divergentseries #divergent #convergent #geometricseries #abelsum #somepi

[hughobyrne]

"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.

[Tejas-id]

This man can do visual proof of anything

[abhinavanand]

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.

[feynstein]

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 😂😂😂

[binbots]

Math is just counting an infinite amount zeros.

[nordicexile]

You also get -1/12 when integrating the equation y = x(x+1)/2 evaluating over [-1, 0]. That's the formula for summing the natural numbers, but using a continuous curve rather than the discrete values in the normal N summation. I don't think that's a coincidence, but I don't know how to prove otherwise.

(You can see for yourself by using a query of "integrate x(x+1)/2 from -1 to 0" in wolfram alpha.)

[PokeGus]

6:37-6:46 Are you allowed to do this? My maths is rusty but I vaguely recall that changing the order of terms in an infinite summation changes the value it limits to.
(Or maybe that's me digging into the whole reason this maths is weird 😂😂)

[MegaLokopo]

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.

[vitowidjojo]

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.

[SurfTheSkyline]

I had no idea how you'd pull it off but this is beautifully and elegantly done! Incredible work!

[claudiocosta]

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

[themightyripples]

This was freakin awesome and deserves more views! This is now my favorite way of trying to explain this to someone - especially those who don't understand math so well

[CasualGraph]

so it's just the numberphile argument but with more squares

[tomkerruish]

This reminds me of the (in)famous Numberphile video on this topic, except that you actually give some (admittedly handwavy) justification to say Grandi's series sums to ½. Subscribed!

[DavyCDiamondback]

The first bad math was trusting the limit from one side. The second was assuming you could slide the columns of the square grid into the infinite triangle without altering that sum. Pretty sure there's a flaw in the S-N=4S argument too, but not as sure, maybe I'm just sleepy

[eknight]

Now I wonder if there is a construction that would make us identify 12S as 1 hollow square.

[akultechz]

1:01 When You do S - N. You add 2 for subtracting 1 from S, similarly, add 4 for removing 3 and so on.. which means S - N = S + x where x = 1×(number of total terms)
Which is not equal to 4S because the number of terms are still same in both S and N so the operation S - N can not increase it by additional threefold

[mustafaseyitt]

It's funny when people play around divergent sums. Triggers me everytime man.

[marcevanstein]

It's interesting: the alternating sum 1 - 1 + 1 - 1 + 1 - 1 + 1 doesn't converge as a sequence of partial sums. But I guess the idea here is that it does converge as a sequence of infinite series 1 - r^2 + r^3 - r^4 .... if you let r approach 1 from the left. It's like, depending on the order / approach of trying to make the series converge, it either does or doesn't converge. I'm not sure I'm articulating this well.

[superbfacts]

but isn't the limit not the real value? a limit is just a limit?