1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Uncountable Sum

  1. Aug 3, 2009 #1
    Does there exist a converging uncountable sum of strictly positive reals?
     
  2. jcsd
  3. Aug 4, 2009 #2
    Would that be an integral?
     
  4. Aug 4, 2009 #3
    No. I mean an actual uncountable sum. An (Riemann) integral is the limit of a sequence of countable sums.
     
  5. Aug 4, 2009 #4
    In the sensible way to define uncountable sums, you prove that for a sum of real terms, if it converges (to a real number) then all but countably many terms must be zero.
     
  6. Aug 4, 2009 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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.
     
  7. Aug 4, 2009 #6
    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] .
     
  8. Aug 4, 2009 #7
    I'm surprised. I thought this would have been defined at some point.

    Suppose [tex]x_i[/tex] is a (possibly uncountable) sequence of positive reals indexed by some ordinal L. Then their sum is [tex]\sum_{i\in L}x_i=\sup\{\sum_{i\in k}x_i:k<L\}[/tex]. This takes care of limit ordinals.

    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?
     
  9. Aug 4, 2009 #8
    I don't know what this definition is trying to achieve. I prefer mine.
     
  10. Aug 5, 2009 #9
    If and only if [itex]D[/itex] is countable.
     
  11. Aug 5, 2009 #10
    Well [tex]\epsilon_0[/tex] is the first uncountable ordinal, so why not?
     
  12. Aug 5, 2009 #11
    Strange ... [itex]\epsilon_0[/itex] is commonly used to represent a certain countable ordinal, while [itex]\omega_1[/itex] denotes the least uncountable ordinal. In any case, that notation doesn't matter. Here is a repeat of the same answer as before: If a sum of positive real terms converges to a finite value, then the index set is countable.
     
  13. Aug 5, 2009 #12
    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.
     
  14. Aug 6, 2009 #13
    Hint for the proof: the real line has a countable dense set, and every term of the convergent series corresponds to an interval.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook