Vector Space spanned by the polynomials $p_n(x) = x^n$ 

Let $V$ be a vector Space Spanned by the set $\mathbb B = \{ p_n(x) = x^n | x \in \mathbb R , n \in \mathbb N \}$. Is $V$ is a vector Space of all real valued continous function on $\mathbb R$


This question is given in my question paper. I think $V$ is a vector space of all real valued continous function on $\mathbb R$, because every continous function on $f$ on $\mathbb R$ can be written as $f(x) = \sum_{n=1}^{\infty} a_n x^n $, where $ a_n , x \in \mathbb R$. But in answer key it is a proper subspace of a vector space of all real valued continous function on $\mathbb R$. So please tell me what is the right answer.
Thank you.
 A: Take a set $B$. The vector space spanned by $B$ is the set of all finite sums $\sum_{k=1}^n \lambda_k a_k$ with $a_k \in B$. So the vector space spanned by $\mathbb B$ is the set of all polynomials (because you only have finite linear combiniations of functions $x^n$)
.
Update: Please note, that not each continuous function $f$ can be represented as $f(x)=\sum_{n=1}^\infty a_n x^n$. Take $$f(x)=\begin{cases} \exp\left(-\frac 1{x^2}\right) & ; x\neq 0 \\ 0 & ;x=0\end{cases}$$
as a counterexample.
A: In complex analysis you will meet some functions which are not possible to express as polynomials or monomial series but can be if we allow negative exponents too. i.e. Laurent expansions around some point $z_0$: $$\sum_{k=-\infty}^{\infty}{(z-z_0)}^k$$
These can be used to describe neighbourhoods of singularities, such as the point 0 in $\exp(-1/z)$. For many simple functions this becomes in practice the same as doing a substitution into the Taylor expansion to get also negative exponents, but the theory is much richer.
As many have already mentioned there is an important distinction between finite dimensional and infinite dimensional vector space. You will learn more about this if you study "functional analysis". 
