Question: Is it true that if the taylor series of $f$ converges it converges to $f$?

So if I want to prove that $\lim R\_n(x)=0$, could I prove this using the fact that a given taylor series converges ?

Answer: 
No, convergence of the taylor series does not imply convergence of the taylor series to $f$. Consider for example $$f(x) := \begin{cases} e^{-\frac{1}{x}} & x > 0 \\ 0 & x \leq 0 \end{cases}$$ $f$ is smooth, $f^{(k)}(0)=0$ for all $k \in \mathbb{N}$. Thus $$0 = \sum\_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!} \cdot (x-0)^k \not= f(x)$$ for $x>0$.
A function is defined to be real analytic if it equals its Taylor series. Have a look at non-analytic functions to see where this might not be the case. There is an example of a smooth function which has all derivatives $0$ at a point but it isn't a constant function.