Summing divergences and the real projective line

[japplepie]

The real projective line states that there is not difference between positive and negative infinity (maybe except the path needed to be taken to "reach" either one of them) and they are actually connected.

There are a lot of ways to get a definite sum from a divergent series; one of which is the algebraic way (my personal favorite).

note:j(x)=x^0+x^1+x^2+x^3+...
Using the algebraic method I could derive the sum of j(2) as shown below:
j(2)=1+2+4+8+16+...
j(2)=1+2j(2)
j(2)= -1= 1+2+4+8+16+...

Does this mean that j(2) diverges so much that it went pass infinity from the positive side and landed on a definite negative point?
------------------------------------------------------------------------------------------
If the statement above is true, then the idea of < and > falls apart.

Example:j(2)+j(1/2)-1=0 ( the extra -1 here is to account for the (1/2)^0 )
A sum of all positive numbers that equals 0.

A notion of < and > that would work in this scenario is by measuring how far it "moved" in the RPL.
j(2)+j(1/2)-1 "moved" and circled back to 0, but 0 never "moved".
------------------------------------------------------------------------------------------
j(4)=1+4+16+64+256+...
j(2)=1+2+4+8+16+32+64+128+256...=(1+2)+(4+8)+(16+32)+(64+128)+...
j(2)=3+12+48+192+...=3(1+4+16+64+...)
j(2)=3j(4)

S2:={1,2,4,8,16,32,64,128,...}
S4:={1,4,16,64,256,...}
S2\S4={2,8,32,128,...} which is S4 multiplied by 2 element-wise
S2-S4=2(S4) (treat it like a sum)
S2=3(S4)

Does this also mean that in j(x), the smaller x (x>=1), the bigger* it is.
Bigger* meaning it "moved" more.

 

[Office_Shredder]

We actually had a similar thread here already not too long ago

https://www.physicsforums.com/showthread.php?t=727863

Basically the moral of the story is that you shouldn't take the results of manipulating divergent series too seriously.

 

[gopher_p]

The real projective line states that there is not difference between positive and negative infinity (maybe except the path needed to be taken to "reach" either one of them) and they are actually connected.

The real projective line is the one point compactification of the real line. It is a topological space, not a sentient being with the ability to communicate. It does not state anything.

In this space, there is no "positive" infinity or "negative" infinity. There are not two points to differentiate. There is one new point. Saying that this point is different from itself in some way is ridiculous.

There are a lot of ways to get a definite sum from a divergent series; one of which is the algebraic way (my personal favorite).

note:j(x)=x^0+x^1+x^2+x^3+...
Using the algebraic method I could derive the sum of j(2) as shown below:
j(2)=1+2+4+8+16+...
j(2)=1+2j(2)
j(2)= -1= 1+2+4+8+16+...

Does this mean that j(2) diverges so much that it went pass infinity from the positive side and landed on a definite negative point?

This method assumes a lot of things about infinite sums; namely that this one in particular is a number and that all of the algebraic properties of finite addition and multiplication can be extended to infinite addition, whatever the hell that is. Since you haven't stipulated what infinite addition is nor why we should accept the aforementioned assumptions, there is no way to tell what your result "means".

If the statement above is true, then the idea of < and > falls apart.

Why? There are two very obvious linear orderings of the real projective line which extend the usual ordering on the reals.

Maybe you mean that this new ordering doesn't "play with" addition on the real projective line the same way the old ordering did on the real line. But regardless of where you put ∞ in this ordering, under the usual assignment of a+∞=∞ the theorems involving order and addition are still true.

So this is really about how your infinite addition interacts with the ordering. Again, until you tell us what infinite addition is, we can't confirm any observation regarding how the order is broken.

Example:j(2)+j(1/2)-1=0 ( the extra -1 here is to account for the (1/2)^0 )
A sum of all positive numbers that equals 0.

A notion of < and > that would work in this scenario is by measuring how far it "moved" in the RPL.
j(2)+j(1/2)-1 "moved" and circled back to 0, but 0 never "moved".
------------------------------------------------------------------------------------------
j(4)=1+4+16+64+256+...
j(2)=1+2+4+8+16+32+64+128+256...=(1+2)+(4+8)+(16+32)+(64+128)+...
j(2)=3+12+48+192+...=3(1+4+16+64+...)
j(2)=3j(4)

S2:={1,2,4,8,16,32,64,128,...}
S4:={1,4,16,64,256,...}
S2\S4={2,8,32,128,...} which is S4 multiplied by 2 element-wise
S2-S4=2(S4) (treat it like a sum)
S2=3(S4)

Does this also mean that in j(x), the smaller x (x>=1), the bigger* it is.
Bigger* meaning it "moved" more.

 

[japplepie]

The real projective line is the one point compactification of the real line. It is a topological space, not a sentient being with the ability to communicate. It does not state anything.

It means that the idea behind it states ...





This method assumes a lot of things about infinite sums; namely that this one in particular is a number and that all of the algebraic properties of finite addition and multiplication can be extended to infinite addition, whatever the hell that is. Since you haven't stipulated what infinite addition is nor why we should accept the aforementioned assumptions, there is no way to tell what your result "means".


And yes infinite operations are still very shady.

 

[CellCoree]

I find the concept of summing divergences and the real projective line to be fascinating. The idea that there is no difference between positive and negative infinity and that they are connected challenges our traditional understanding of infinity. It also raises interesting questions about how we define and measure infinity.

The algebraic method you have used to derive the sum of j(2) is a valid approach, but it is important to note that it is not the only method. Different methods may yield different results, and it is up to us as scientists to critically evaluate and compare these different approaches.

The examples you have provided also highlight the need for caution when dealing with divergent series. As you have shown, seemingly contradictory results can arise when manipulating these series. This emphasizes the importance of clearly defining our terms and being mindful of the assumptions and limitations of our methods.

In regards to your last question, I would caution against making a blanket statement that smaller x in j(x) always leads to a "bigger" result. It may be true in some cases, but it is not a general rule. It is important to carefully consider the behavior of each individual series and how it relates to the concept of infinity.

Overall, the concept of summing divergences and the real projective line is a complex and thought-provoking topic. As scientists, it is our responsibility to continue exploring and questioning these ideas in order to deepen our understanding of infinity and its implications in mathematics and beyond.