I have read somewhere that we can extend the notion of a series of a sequence(adsbygoogle = window.adsbygoogle || []).push({});

[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 cant 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 wich I made the sum?

Thanks in advance for any help in understanding this.

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# Sums over arbitrary index set

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