You can just pretend that ω is finite and plug it into the formula for the partial sum.n∑i=1i=12n2+12n, so ω∑i=1i=12ω2+12ω. If they were to give the ith odd number amount of fish on the ith day (1,3,5,7,9...), then you would have ω2 amount of fish, because n∑i=12i−1=n2. The two links I posted about the handling of infinite divergent series go into greater detail (eg. the question of the starting index).
The links are very on point for my interest thanks for those. Some of it is in rather dense math but alas that is the case when the topic is math.
At one point there is a constuction where in addition to having series of real numbers to define a hyperreal (r1,r2,r3...)=h1 we define a series of hyperreals (h1,h2,h3...)=d1, in order to get a “second tier hyperreal”. So I do wonder whether the “fish gotten per day” is adeqate to distinguish between the scenarios. That is there might be a difference between “each day I get promised an infinite amout of fish” and “each day I get 1 more fish”. That is on day n I have been promised ωn fish and taking it as α∑i=1I am not sure whether α=ω and whether terms like ω2 and ωα refer to the same thing or whether mixing “first-level” and “second level” hyperreals gets you a thing different than mixing just “level 1”s
You can absolutely count your fish that way with the help of hyperreals! (“growing promise” stream would be 12ω2+12ω though)
I think https://en.wikipedia.org/wiki/Hyperreal_number#The_ultrapower_construction is a good introduction. https://math.stackexchange.com/questions/2649573/how-are-infinite-sums-in-nonstandard-analysis-defined and https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3459243 address the handling of infinite divergent series with hyperreals and https://arxiv.org/pdf/1106.1524v1.pdf talks about uniform probability over N (among other things).
Why 12ω2+12ω and not any other? What kind of stream would correspond to ω2 ?
You can just pretend that ω is finite and plug it into the formula for the partial sum.n∑i=1i=12n2+12n, so ω∑i=1i=12ω2+12ω. If they were to give the ith odd number amount of fish on the ith day (1,3,5,7,9...), then you would have ω2 amount of fish, because n∑i=12i−1=n2. The two links I posted about the handling of infinite divergent series go into greater detail (eg. the question of the starting index).
The links are very on point for my interest thanks for those. Some of it is in rather dense math but alas that is the case when the topic is math.
At one point there is a constuction where in addition to having series of real numbers to define a hyperreal (r1,r2,r3...)=h1 we define a series of hyperreals (h1,h2,h3...)=d1, in order to get a “second tier hyperreal”. So I do wonder whether the “fish gotten per day” is adeqate to distinguish between the scenarios. That is there might be a difference between “each day I get promised an infinite amout of fish” and “each day I get 1 more fish”. That is on day n I have been promised ωn fish and taking it as α∑i=1I am not sure whether α=ω and whether terms like ω2 and ωα refer to the same thing or whether mixing “first-level” and “second level” hyperreals gets you a thing different than mixing just “level 1”s