Fields closed under countable additions?

Question

So I was learning about fields today and I came across the property that fields are closed under addition. That is, if $v_2,v_1 \in $ F ,then:
$ v_1 + v_2 \in$ F
Similarly, I we can say that:
$ v_1 + v_2 +v_3\in$ F
My question is can we extend this by saying:
$ \sum_{n=1}^{\infty} v_n \in$ F ?
Where $v_n \in $ F
I am pretty sure it won't hold for uncountably infinite additions, but will this be true for countably infinite addition as shown?

Answer 1

No. Consider the field $\mathbb R$, and the countable sum $\sum\limits_{n=1}^{\infty}1$.

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Additional comments:

The problem is that an infinite "sum" is not a sum at all. It is definitely NOT an infinite number of repeated additions. It is the limit of the sequence of finite sums (each of which IS an honest-to-goodness sum, and is an element of the field).

For your sum to exist in an arbitrary field, you would need to be able to define such a limit as a member of that field.

In the counterexample I gave, the partial sums are just $S_k = \overbrace{1+1+\cdots + 1}^{\textrm{$k$ terms}} = k$, but $\lim\limits_{k\to\infty}S_k$ does not exist as an element of $\mathbb R$.