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Dragonfall
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Does there exist a converging uncountable sum of strictly positive reals?
First you will have to define what you mean by "uncountable sum"! I know a definition for finite sums and I know a definition for countable sums (the limit of the partial, finite, sums), but I do not know any definition for an uncountable sum except, possibly the integral that bpet suggested.Dragonfall said:Does there exist a converging uncountable sum of strictly positive reals?
g_edgar said:Definition Let [itex]S[/itex] be an index set. Let [itex]a \colon S \to \mathbb{R}[/itex] be a real function on [itex]S[/itex]. Let [itex]V[/itex] be a real number. Then we say [itex]V = \sum_{s\in S} a(s)[/itex] iff for every [itex]\epsilon > 0[/itex] there is a finite set [itex]A_\epsilon \subseteq S[/itex] such that for all finite sets [itex]A[/itex] , if [itex]A_\epsilon \subseteq A \subseteq S[/itex] we have [itex]\left|V - \sum_{s \in A} a(s)\right| < \epsilon[/itex] .
Dragonfall said:So does there exist sequences [tex]x_i[/tex] indexed by ordinals [tex]D\geq\epsilon_0[/tex] such that [tex]\sum_{i\in D}x_i[/tex] is finite, and that each x_i is positive?
Dragonfall said:Yes I was mistaken on the notation, it should be [tex]\omega_1[/tex].
You have yet to say why. You asserting it true doesn't constitute a proof.
A converging uncountable sum of positive reals refers to the concept in mathematics where an infinite sum of positive real numbers is considered to converge, or have a finite value, even though the sum contains an uncountable number of terms. This is in contrast to a converging countable sum, where the sum contains a countable number of terms.
The main difference between a converging uncountable sum of positive reals and a converging countable sum is the number of terms involved. A converging uncountable sum has an uncountable number of terms, while a converging countable sum has a countable number of terms. Additionally, a converging uncountable sum involves the sum of positive real numbers, while a converging countable sum can involve any type of number.
No, a converging uncountable sum of positive reals cannot be infinite. By definition, a converging sum has a finite value, meaning it cannot be infinite. However, it is possible for a converging uncountable sum to be equal to the infinity symbol (∞), but this is not the same as being infinite.
An example of a converging uncountable sum of positive reals is the sum of all positive real numbers less than 1. This sum is equal to 1, even though it contains an uncountable number of terms. Another example is the sum of all positive real numbers less than or equal to 1, which is equal to 2.
The convergence of an uncountable sum is determined by using mathematical concepts such as the Monotone Convergence Theorem and the Cauchy Criterion. These concepts help determine if the sum has a finite value, or if it diverges (i.e. has an infinite value or does not have a defined value).