The area in the negative Y axis is -1/12?

@Splashee Yes. To be clear, it's the area in red here:

When you take the integral, the area below the x axis counts as negative, hence the negative value of the result.

@Silly Druid Kinda off topic but still, if I gave you a list of number in sequence, like these:

7
6
7
7
7
7
8
8
9
10
11
13
14
18
22
30
46
84
482

Would you be able to find a close match making a quadratic function graph?

These values are used to create a bend/curve in an old racing game, and the way the numbers were generated are unknown. The 7 to 6 to 7 looks like a rounding to integer situation, meaning the original numbers were real numbers. The steps between them don't need to be exact, but the curve they generate have a particular look when transformed into a pseudo 3D view.

@Splashee This doesn't look like a quadratic, the growth from 84 to 482 is way too fast compared to the earlier steps. Unless the x values aren't evenly distributed.

@Splashee you could try to solve the quadratic equation by the substituting ordered pairs into quadratic equation ax^2  + bx + c = y

@Silly Druid in this case 1+2+3+4+... = -1/12 is true only because "=" doesn't mean "equals to" it means "is associated with" . It's just another way for the mathematicians to lie to others and still be correct.

On 3/7/2023 at 12:00 PM, Silly Druid said:

the growth from 84 to 482

Yes, I have noticed that. Same with all the different curve data. I have to try to find as much info as possible to try to solve what they were trying to do. I am quite sure it has to do with 3D perspective, making things farther away get smaller quicker (larger values in this case means the curve bends quicker, acting like you see it father away in the distance, maybe)

15 hours ago, RDDash said:

It's just another way for the mathematicians to lie to others and still be correct.

I so want to agree with this (in general about mathematicians).

It's like what if I say that 1+2+3+4+5+6+ .... We say that it's represent the value of S.

We also say that 1+ (2+3+4) +(5+6+7) +(8+9+10) + ... Represent the value of S

Now we get that 1 + 9 +'18 ,+27 + 36 ... Is the value of S

That the same as saying as 1+ 9(1 +2 +3+4+5+6+....) Because I rewrote it that way would that still represent the value of S?

We would have 1+9(S) is S. (Or just let's call it a P or something )

 So that would mean that 1 is the same as -8S

By that logic S has the value of -1/8

If we rewrite the sum of value of S as 1+2 + 5(1 +2+3+4+5+6...) We would still end up with 3+5(S) is S, and S = -1/8.

2 minutes ago, RDDash said:

Is the value of S

So S in this case is ∞?

There are many ways to reach ∞, but like with the divide by 0 situation (on the positive side of the number line), 1/0 = ∞ and 2/0 = ∞, so 1=2.

5 minutes ago, RDDash said:

So that would mean that 1 is the same as -8S

I didn't see a sign switch. Addition and multiplication cannot change the sign of a number.

I am probably missing the point here. It is too complex for me.

@Splashee

The sign switch is due to moving the term to the other side of the equation.

S = 1 + 9S

(move 9S to the left side)

S - 9S = 1

-8 S = 1

(divide both sides by -8)

S = -1/8

@RDDash

Such manipulations of divergent infinite sums are normally not allowed and can lead to nonsensical results. It's interesting that your approach leads to the same result in both cases, but I can give you another one that leads to another result:

S = (1+2) + (3+4) + (5+6) + ...

S = 3 + 7 + 11 + ...

but also

S = 1 + (2+3) + (4+5) + (6+7) + ...

S = 1 + 5 + 9 + 13 + ...

now let's add both expressions for S

2S = 1 + (3+5) + (7+9) + (11+13) + ...

2S = 1 + 8 + 16 + 24 + ...

2S = 1 + 8S

-6S = 1

S = -1/6

Also this kind of manipulations were used in the Numberphile video, without explaining properly why they were "allowed" in that case, which makes that video a bad example of popularizing mathematics. (Watch the Mathologer video that I linked in the main post for more information about that.)