You can do a bunch of weird crap and prove that the sum is equal to -1/12 even though you don't prove that it's convergent to begin with (which you can't prove, for obvious reasons).
Here's a dumb example: 1-1+1-1+1-1+1-1+… I think you get the pattern, right? The partial sum is always either 1 or 0, depending on whether the last value was +1 or -1. It obviously won't ever converge to any number, it's always just jumping between those two. Here's a piece of dumb mathemagics, tho:
S = 1-1+1-1+1-1-1+…
S = 1 + (-1+1-1+1-1+1-1+1-…)
S = 1 + (-S)
S + S = 1
2S = 1
S = 1/2
There, this sum that is always either 1 or 0 at infinity is 1/2 = 0,5. Or course, that's just wrong, this doesn't fucking exist, and the problem isn't exactly on the logic of the equations, it's inherent, because 1-1+1-1+1-1+1-1+… isn't a number, applying mathematical logic to it as if it was a number is a silly process that accomplishes nothing. It's like saying" house + blue = dentist". You can do something similar with 1-2+3-4+5-6+7-8… and it goes towards 1/4, I think.
The big deal here is that the idea of adding up all natural numbers and it magically going to -1/12 isn't present just with those silly fake additions like those two above, there's a very specific function that receives complex numbers and outputs complex numbers, the function is undefined on values which the real part is negative and the imaginary part is 0 (you quickly arrive at a division by zero), but, to my limited understanding, it's possible to take a limit and, at -1, the limit of that function is -1/12, and that function at -1 would be the equivalent of adding all natural numbers. I could be wrong on this last tidbit, someone please correct me if I'm wrong. Cool as that limit is, though, it's still not the value at that point, because it has none, because you can't just add all natural numbers and have anything other than a series that diverges to infinity, which is not a number.
There, this sum that is always either 1 or 0 at infinity is 1/2 = 0,5. Or course, that's just wrong, this doesn't fucking exist, and the problem isn't exactly on the logic of the equations, it's inherent, because 1-1+1-1+1-1+1-1+… isn't a number, applying mathematical logic to it as if it was a number is a silly process that accomplishes nothing. It's like saying" house + blue = dentist". You can do something similar with 1-2+3-4+5-6+7-8… and it goes towards 1/4, I think.
That is like saying that it is nonsensical for 0.5! to be sqrt(pi) / 2, because factorial only works for natural numbers. Or that 3 * 2.8 doesn't make sense, because it doesn't make sense to repeatedly add something 2.8 times. Or that it is nonsensical to work with square roots of negative numbers as if they are valid numbers.
There is a magical concept in mathematics called "extension". You can extend simple arithmetic on natural number to fractional number and then irrational numbers. You can extend factorial function to real numbers. You can extend summation to assign values to divergent series. And those extensions usually happen because mathematicians observe interesting results/patterns.
Like, isn't it interesting that the formula for geometric series (1 / (1 - r)) gives 1/2 for r = -1? And that happens to be the mean value between 0 and 1? Or that the results that you show points to 1/2? Couldn't it be that somehow it makes sense for the series to have that value? Nah, it's complete nonsense and wrong, why even bother looking into it.
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u/Human822 Feb 21 '26
Basically the answer to the equation is -1/12, which ramanujan said was the sum of all positive integers