You might have already heard this ‘fact’ somewhere. My city even had the equation as a part of its Christmas decoration (lighted up and everything). It turns out what we got for Christmas was lies.
By our definitions of limits (and everything else), the sum equals infinity. As you’d expect. Regardless, there are some connections between the sum and -1/12. They might look like proofs to an untrained eye, but are really just connections.
Proof one
Let’s define the following sum
A = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + …
and look at its partial sums. If we add the first two terms, we get 0, then we add 1 and get 1, then it’s 0 again, then 1, … So, this infinite sum is the limit of the sequence
1, 0, 1, 0, 1, 0, 1, 0, 1, …
What would be the most natural way to define this limit? Taking averages gets us
1, 1 / 2, 2 / 3, 2 / 4, 3 / 5, 3 / 6, …
This sequence converges to ½, so I guess it would make sense to say that our initial sequence limits to ½, meaning A = ½.
For the sum
B = 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + …
we can notice that
2 B =
= 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + …
+ 0 + 1 - 2 + 3 - 4 + 5 - 6 + 7 - … =
= 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + … =
= A
So B = A / 2 = 1 / 4.
If we take our sum of all natural numbers
S = 1 + 2 + 3 + 4 + 5 + 6 + …
then
S - B =
= 1 + 2 + 3 + 4 + 5 + 6 + … -
- 1 + 2 - 3 + 4 - 5 + 6 - … =
= 0 + 4 + 0 + 8 + 0 + 12 + … =
= 4(1 + 2 + 3 + …) = 4 S.
We got 3S = -B = -1/4, which gives
S = - 1 / 12.
So, if your friend out of nowhere decides to one day give you a dollar and the next day two, then three, … End that toxic relationship and sue him for stealing. Use the above proof in court.
But seriously, these kinds of manipulations with divergent sums are forbidden in math, as you can quickly get contradictions. E.g.
E = 1 + 1 + 1 + 1 + 1 + 1 + …
0 = E - E =
= 1 + 1 + 1 + 1 + 1 + 1 + 1 + … -
- 0 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - … =
= 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 1.
Apparently, this was proof zero.
Proof two
This one uses some complex analysis. Maybe you have heard of the Riemann zeta function. It’s a pretty important function and finding its zeroes is a problem worth a million bucks. Its definition is
where s can also be a complex number. The thing with complex functions is that they can sometimes be incredibly nice. So nice, that knowing their values inside any disk on the complex plane, gives us exactly one natural extension on some larger area, like e.g. the whole complex plane.
The Riemann zeta function is really nice when the real part of s is larger than 1. This gives us enough information to extend it to the whole complex plane. This is called analytic continuation. And it turns out that this continuation evaluated at s = -1 gives us -1/12.
We formally can’t just plug s = - 1 in the sum on the right as that notation does not preserve analytic continuation. But it is very interesting that in this complex world, -1/12 is the only natural definition for the sum of all natural numbers.
Conclusion
Don’t believe anyone who tries to convince you that any of the above arguments is actual proof (unless his math has some freaky definitions). Though if you are disappointed, math has a bunch of real paradoxes that may be as strange as this one.
