Taylor expansion of smooth function with compact support

Question

If one has a function $\varphi(x) \in C_c^{\infty}(\mathbb{R})$, it is not analytic (except for in the trivial case $\varphi(x)=0$), so in the Taylor expansion $\varphi(x) = \sum_{k=0}^{n} \frac{\varphi^{(k)}(x_0)(x-x_0)^k}{k!} +R_n(x)$ the remainder doesn't go to zero as $n$ increases. Can one say more about the Taylor expansion? Does it hold that $R_n(x) \in C_c^{\infty}(\mathbb{R})$? Forgive me if the question is stupid - one still has that $\bigg( \sum_{k=0}^{n} \frac{\varphi^{(k)}(x_0)(x-x_0)^k}{k!} +R_n(x) \bigg) \in C_c^{\infty}(\mathbb{R})$?