1+2+3+4+....infinity = -1/12?

[Enola_Straight]

last chance bump.

Telemark?
Ultrafilter?

[jezzaOZ]

What I find extraordinary, and no-one has even mentioned it, is that the sum of 1,2,3,4… is equal to exactly twelve times e^(i.pi) .

Spooky eh?

[Great_Antibob]

@jezzaOZ

What I find extraordinary, and no-one has even mentioned it, is that the sum of 1,2,3,4… is equal to exactly twelve times e^(i.pi) .

Spooky eh?

1. How so? e^(pi*i) = -1. Twelve times this is -12. Did you mean 1/12 instead?

2. Why is this spooky? That’s no more spooky than (3i)^2 being 9 times bigger than e^(pi*i). They’re two numbers for which a whole number ratio can be found. There are literally infinitely many numbers for which such a relation can be found.

ETA (3), of course, whatever equality was meant, it only exists when that particular series goes to -1/12, which is true for a particular notion of infinite summation and not generally true.

[Great_Antibob]

@Enola_Straight

Suppose we work from the opposite direction:

Does -1-2-3-4…=+1/12?

In some sense, this is true.

You can pretty much approach it as the negative of the original sum. In the same sense that 1+2+3+4+… = -1/12 can be said to be true, you can say that -1-2-3-4-… = 1/12.

[Indistinguishable]

@Enola_Straight

Suppose we look at this problem as part of the Complex Plane:
Complex plane - Wikipedia

and describe a Ray…with and endpoint a the Origin 0 and extending out to infinity.

As we sweep the ray around, describing i+2i+3i+4i…, -i-2i-3i-4i…, in fact ALL “a+bi”. would the origin have a perfect circle of r=1/12?

Or perhaps some irregular shape like a fractal?

I am unable to make sense of this.

For example, 0 + 1 + 2 + 3 + 4 = 10, but if I take a ray from 0 to 4, and sweep it around the origin in the complex plane, I don’t get a circle of radius 10. I get a circle of radius 4.

I suspect your thinking here was muddled (but, of course, there is some possibility that you may demonstrate that it is actually a confusion on my part in misinterpreting you).

[Chronos]

To clarify, I think he’s asking if 1j+2j+3j+4j+… = -1/12 j, for all j such that |j| = 1. Thus, if you sweep the original series around the origin, the sum of the series would form a circle about the origin with a radius of 1/12.

[Indistinguishable]

@Chronos

To clarify, I think he’s asking if 1j+2j+3j+4j+… = -1/12 j, for all j such that |j| = 1.

Ah. This is true for all j whatsoever, on our methodology; scaling a series by a factor scales its sum by the same factor, as expected.

Thus, if you sweep the original series around the origin, the sum of the series would form a circle about the origin with a radius of 1/12.

Oh, I see. Sure, although let’s note that the rotated series in almost any direction would not consist of precisely the values a + bi for integer a and b in that direction, since in almost any direction, the corresponding unit vector would not have integer components. Rather, in almost every direction, the corresponding rotation of our original series would include some non-integral values and exclude all the integral values. (In directions corresponding to integer Pythagorean triples, the corresponding series would include all the integral values, but still, in almost all cases, also include further non-integral values. Just in four particular directions (+1, -1, +i, -i) will the series match up with the integral values.)