- #1
Damidami
- 94
- 0
I have read somewhere that we can extend the notion of a series of a sequence
[tex] \sum_{i=1}^{\infty} a_n [/tex]
to sums over an arbitrary index set, say
[tex] a : I \to \mathbb{R} [/tex]
is a family of real number indexed by I, then
[tex] \sum_{i \in I} a_i [/tex]
is the sum of all the elements.
I think the text said that [tex]a_i \geq 0 [/tex] in order to has sense, and that if [tex]a_i \neq 0[/tex] for a non-numerable size of elements, then the series can't converge.
1) My question is with that last sentence, why can't converge such a sequence?, for example if the family is
[tex]a : \mathbb{R_{>0}} \to \mathbb{R}, a(i) = \frac{1}{i}[/tex]
how does one sum over all it's elements?
2) Other question, the fact that all the [tex]a_i \geq 0[/tex] is required because we can't asume an order on the elements of the index set, and if I have negative elements the convergence can vary according to the order in which I made the sum?
Thanks in advance for any help in understanding this.
[tex] \sum_{i=1}^{\infty} a_n [/tex]
to sums over an arbitrary index set, say
[tex] a : I \to \mathbb{R} [/tex]
is a family of real number indexed by I, then
[tex] \sum_{i \in I} a_i [/tex]
is the sum of all the elements.
I think the text said that [tex]a_i \geq 0 [/tex] in order to has sense, and that if [tex]a_i \neq 0[/tex] for a non-numerable size of elements, then the series can't converge.
1) My question is with that last sentence, why can't converge such a sequence?, for example if the family is
[tex]a : \mathbb{R_{>0}} \to \mathbb{R}, a(i) = \frac{1}{i}[/tex]
how does one sum over all it's elements?
2) Other question, the fact that all the [tex]a_i \geq 0[/tex] is required because we can't asume an order on the elements of the index set, and if I have negative elements the convergence can vary according to the order in which I made the sum?
Thanks in advance for any help in understanding this.