Let $V$ be the vector space of all functions $f : \mathbb{R} \to \mathbb{R}$. Recall that the Taylor series for $e^{x}$ centered at $0$ is $$e^{x} = \displaystyle 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots.$$ Does this mean that $e^{x}$ is a linear combination of the vectors in the set $\lbrace 1, x, x^2, x^3, x^4, \ldots \rbrace$?

Explain your answer using complete sentences with correct grammar, spelling and punctuation.