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 real problem is that infinite sums should be understood as some sort of limit of a converging sequence.  But while that operation may make sense, it isn't a vector space operation.
If you add a topology, then you can introduce infinite sums and things will make perfect sense (as long as the sum converges).  In this case you could start with open balls consisting of open interval in the leading dimensions combined with any possible value in the rest.  Now all the infinite sums you want work perfectly - as long as there is pointwise convergence in each dimension the infinite sum is defined.
But there are a lot of topologies you can use.  For example maybe your basic open balls are open intervals of length $l$ in all dimensions.  Now infinite sums are defined so long as they converge uniformly in all dimensions.  Now the sums that you think should work, don't.
Changing the topology will leave the vector space properties alone, but infinite sums will work out differently.
So long and short is this.  Lots of mathematical structures are infinite dimensional vector spaces with infinite sums.  (An important example you'll learn is Hilbert spaces.)  But infinite sum is never a vector space operation.
