$0+0+0+… $ in $(ℝ, +)$ Consider the group $(ℝ, +)$ under ordinary addition. Then what is $0+0+0+...$
My conscience says that it is undefined unless a metric is defined.
Is it that the convergence of a sequence or series is talked only in the metric space?
 A: Yes, infinite sums are defined as the limit of the partial sums $\sum_{i=1}^n a_i$, and limits only make sense if there is a metric. In this case the limit of the partial sums is equal to $0$ no matter what metric you put on $\mathbb R$.

Edit: I should add, as Noah Schweber pointed out, that you do not need a metric to define limits. Rather, a topology would suffice, so one can talk about the convergence of infinite sums in any Hausdorff topological group (you need the Hausdorff assumption to ensure limits are unique). For example, in the group $\mathbb R^\mathbb R$ of functions from $\mathbb R$ to $\mathbb R$, it makes sense to talk about infinite sums of functions by considering whether they converge pointwise. This notion of convergence is not induced by a metric, instead coming from the product topology on $\mathbb R^\mathbb R$.  
You do not even need a topology. For example, the notion of almost-everywhere convergence of functions in $\mathbb R^\mathbb R$ is not induced by any topology. The most general setting where you can talk about infinite sums would be an abelian group which is also a Hausdorff convergence space such that the two structures are compatible. 
A: I just wanted to add a trivial addendum to the other answer, and that is that sometimes I find it (notationally) convenient to define infinite sum for any abelian group if all except finitely many terms are $0$.
We can say that $\sum_{i} a_i$ is defined if all but finitely many $a_i = 0$, in which case it is the sum of all nonzero terms.
In your case, $0 + 0 + 0 + \cdots = 0$.
Why would you want to do this? It can be useful and convenient to not worry about summation start/end indices. In any field, for example,
$$
(a + b)^n = \sum_{i \in \mathbb{Z}} \binom{n}{i} a^i b^{n-j}
$$
which is an "infinite sum" where all but finitely many terms are $0$.
Putting it in this form where the sum is over all integers
tends to save a few computational steps when computing nesting sums, switching order of summation, and so on.
