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From Wikipedia, it states that

For an infinite sequence $G_1, G_2, \ldots$ of groups, this can be defined just like the finite direct product of above, with elements of the infinite direct product being infinite tuples.

Now let $\mathbb{Z},\mathbb{Z},\dots$ be an infinite sequence of groups and $G$ be the infinite direct product of $\mathbb{Z}$.
Consider the element $(1,1,\dots)$ in $G$.
Can it be written as an infinite sum of elements of $G$?
That is, $$g=\sum_{i=1}^{\infty}a_i$$ where for a fixed $i$, the $i$-th entry of $a_i$ is $1$ while all the other entries is $0$.

I find this confusing because when we talk about infinite sum/series, we need to consider about limit of partial sums, but it seems that the concept of limit cannot be applied here.

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There is no notion of infinite sum in a group without extra structure. You can sort of write stuff like this down heuristically but you need to be careful when reasoning with it. For example, the image of $g$ under a homomorphism is not determined by the image of the $a_i$. As an extreme example, the quotient $\prod G_i / \bigoplus G_i$ exists and is nonzero iff infinitely many of the $G_i$ are nontrivial.

On the other hand, we can say the following. The discrete groups $G_i$ can be given the discrete topology, and then their infinite product $\prod G_i$ can be given the product topology, which in the infinite case will not be discrete (again iff infinitely many of the $G_i$ are nontrivial). If the $G_i$ are finite then this topology makes the infinite product a profinite group; in general it's only a "prodiscrete" group. A sequence of elements of $\prod G_i$ converges in the product topology iff it converges pointwise, so it is actually meaningful and true to say that an element $g = (g_1, g_2, \dots )$ of the infinite product is the limit, in the product topology, of the sequence of "partial sums"

$$(g_1, e, e, e, \dots)$$ $$(g_1, g_2, e, e, \dots)$$ $$(g_1, g_2, g_3, e, \dots)$$

since this sequence converges pointwise to $g$. Be careful that if you want to use this to conclude anything about homomorphisms out of the infinite product $\prod G_i$ then you need to ask for them to be continuous with respect to the product topology; there are discontinuous ones in general.

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