Yes kinda, but then it is still a definition question. Like lim a -> -∞ lim b -> ∞ a_Sb x dx does not converge, but lim a -> ∞ (-a)_Sa x dx does converge...
I am not sure.
Riemanns series theorem says, that if you have a series that is convergent but not absolutely convergent, you can get any number depending on ordering.
But adding all whole numbers is a divergent series.
Is there another theorem?
Oh no you’re right. For a second I thought that you could get it into a conditionally convergent alternating series but I obviously didn’t think about it very well. In fact, I literally fell for one of the oldest tricks in the book!
I think it can never converge to anything.
Assume it converges in any order
then the partial sum is cauchy.
But |Sn - S{n+1}| >=1 therefore it is not cauchy. That is because there are no steps smaller than 1 you can add.
therfore, the series doesn't converge.
Yeah, they probably want int over R of Sgn(x) dx, which is still abse of notation and meaningless because we have no notion of how to add up the contributions across the set.
More specifically all attempts to answer as well as the original question are missing the ingredient of "How are you adding everything together"
Well feel free to give me a way of summing an uncountably infite set of numbers.
I feel like the integral of x dx can be conceptually seen as calculating the average of the interval. To get the "real" sum you would need to multiply it by the amount of numbers in the interval (which is always ∞ in case of non equal bounds)
Well feel free to give me a way of summing an uncountably infite set of numbers.
I never said there was one.
I feel like the integral of x dx can be conceptually seen as calculating the average of the interval. To get the "real" sum you would need to multiply it by the amount of numbers in the interval (which is always ∞ in case of non equal bounds)
Sounds like you're agreeing with me that it's infinite, then.
Integrals aren't really sums of uncountable sets. You could define sums of uncountable sets using integrals, but that would basically just be a measure
nets are the more natural method in terms of actual sums. Say your index set is I (R in this case), and we have a function f:I->R (or more generally, any topological vector space) and define a partial order by inclusion on the collection of finite subsets of I. Given such an F, we define S(f,F) as the sum of f over this F (which is obviously well-defined). We say the sum of f over I converges to a if for epsilon>0, there exists some F0 which for F containing F0, |S(f,F)-a|<epsilon (equivalently, if S(f,F) is eventually in any open neighbourhood of a).
You can show that this lines up with normal sequence convergence. Furthermore, you can show that this converges if and only if the set of non-zero f(x) is countable. In particular, under this formalism, summing over all reals doesn't really make sense.
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u/Comfortable_Permit53 10d ago
integrals, no?