Sorry for the title, but I think it's funny. Can you write down a homomorphism (of additive groups)

$\mathbb{R}^\mathbb{N} \to \mathbb{R}$,

which is nontrivial and whose kernel contains the finite sequences? For example, on the subgroup of convergent sequences, we can take the limit. The question is not if such thing exists (according to the axiom of choice, $\mathbb{R}^\mathbb{N} / \mathbb{R}^{(\mathbb{N})}$ has a basis over $\mathbb{R}$, etc.). I want to write something down^{1} in order to play around with this "limit for divergent sequences", which might be helpful here. Possibly all of you immediately think that this is not possible, but for which reason? Perhaps it works somehow, but it's just complicated?

^{1}in an informal sense. I'm not interested in a discussion about mathematical logic ;-).