Is $e^x$ in the span of $\{1,x, x^2,...\}$ in the vector space $C[0,1]$ on $[0,1]$? T. Gunn's answer to a question about linear combinations being restricted to finite sums asserted that

For example, you would not count the "infinite" linear combination
$$\exp(x) =  \sum_{n = 0}^\infty \frac{x^n}{n!} $$
to be in the span of $\{1,x,x^2,\dots\}$ in the vector space $C[0,1]$ of continuous functions on $[0, 1]$.

After some brief reading, I can see no obvious reason why a sum of basis vectors should not result in something that is within the span of that basis.
 A: Because an $infinite$ sum is not a sum, it's the limit of a sequence. We calling it "infinite sums" is just abuse of language.
A: In the definition of a basis of a vector space we only permit finite linear combinations of basis vectors.  A general vector space does not give us the concept of convergence, so we cannot define infinite sums by the usual technique of convergence to a limit.
A: If $S = \{v[\lambda] : \lambda \in \Lambda \}$ is an indexed family of vectors (I'm using square brackets to avoid double subscripts) then we define the span of $S$ to be the set of all sums of the form
$$ \sum_{i = 1}^n a_i v[\lambda_i] $$
where $n$ is some finite number, $a_1,\dots,a_n$ are scalars and $\lambda_1,\dots,\lambda_n \in \Lambda$. We do not consider infinite sums because not all infinite sums make sense. For instance
$$ \sum_{i = 0}^\infty x$$
does not make sense.
There are however contexts where you do want to consider infinite sums. To do this you need some notion of convergence. For instance, with respect to the norm topology on $C[0,1]$, the sequence
$$ 1, 1 + x, 1 + x + \frac{x^2}{2}, 1 + x + \frac{x^2}{2} + \frac{x^3}{6}, \dots$$
converges to $\exp(x)$. Each element of this sequence is an element of $\operatorname{span}\{1,x,x^2,x^3,\dots\}$ so their limit is an element of the closure (the set of limit points) of that span: $$\exp(x) \in \overline{\operatorname{span}\{1,x,x^2,x^3,\dots\}}.$$ When dealing with topological vector spaces, it is common to look at the closed span (i.e. the closure of the span).
Without a topology, we cannot make sense of "closure" or "infinite sums" (which are really limits). Since we want to work with vector spaces without a topology on them (i.e. without a notion of limit) we build our definitions without using topological properties.
A: No. You cannot write $\exp(x)$ as a finite linear combination of monomials (i.e. a polynomial) since the $(n+1)$th derivative of a polynomial of degree $n$ is $0$ but $\exp(x)$  is nonzero for all $x$.
