Why does addition not make sense on infinite vectors? I was reading http://www.math.lsa.umich.edu/~kesmith/infinite.pdf to learn more about infinite dimensional vector spaces, and the author argues that the standard basis ($e_i$ is the sequence of all zeroes except in the i-th position, where there appears a 1), does not form a basis for $\mathbb{R}^\infty$ because the span is only defined over the sum over finitely many basis vectors. So, she argues, a vector like $(1,1,1,\dots)$ is not in the span.
If we allow the span to be defined over an infinite sum, then, the author argues, something like $(1,1,1,\dots)+(2,2,2,\dots)+(3,3,3,\dots)$ does not make sense, and thus we have to restrict the span to a finite sum.
I do not understand why this is not simply $(6,6,6,\dots).$ More fundamentally, why can't we generalize the span to include an infinite sum of the basis vectors?

The span is better defined as the smallest (sub)space that contains the given vectors. As the set of all finite linear combinations is a subspace, the span cannot be larger than that.
