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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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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".

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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.

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# Summing divergences and the real projective line

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