Let $f$ be an infinitely differentiable function, and let $T(x) = \sum_{n=0}^{\infty} a_n x^n$ be its Taylor series (say at $x = 0$).
The Taylor series of $f$ need not converge at any point other than $x = 0$. Indeed, by a famous Theorem of Borel, for any sequence $\{a_n\}_{n=0}^{\infty}$ of real numbers whatsoever, there exists an infinitely differentiable function with Taylor series equal to $\sum_{n=0}^{\infty} a_n x^n$. If the $a_n$ grow too rapidly -- e.g. $a_n = n!$ -- then there will be convergence only at $x = 0$.
Even if the Taylor series has a positive radius of convergence $R$, there is no guarantee that $f(x)$ and $T(x)$ must be equal on $(-R,R)$. A function with this property is said to be analytic at $x = 0$. (It should be mentioned that most of the familiar functions one encounters in calculus are analytic.) Perhaps the simplest example of a non-analytic function is $f(x) = e^{\frac{-1}{x^2}}$ (and $f(0) = 0$), for which $f$ and all of its derivatives vanish at $x = 0$, so $T(x) = 0$. But $f(x)$ is clearly positive at any $x \neq 0$.
Added: To show that a function is equal to its Taylor series in some interval $I$, one has to show, for each $x$ in $I$, that the remainder function $R_n(x) = f(x) - T_n(x) = f(x) - \sum_{k=0}^n \frac{f^{(k)}(0)}{k!} x^n$ approaches $0$ as $n$ approaches infinity. The most useful basic tool for this is Taylor's Formula for the Remainder. The issue comes down to having a good understanding of the growth of the derivatives at $0$ as $n$ increases. For instance, if there exists a fixed $M$ such that
$|f^{(n)}(a)| \leq M$ for all $a \in I$, then Taylor's Formula immediately implies that
$R_n(x) \rightarrow 0$ for all $x \in I$. This is the case for instance for $f(x) = \cos x, \sin x, e^x$ (in the last case we need to assume that the interval $I$ is bounded, although we can take it to be bounded as large as we want, so the eventual conclusion will be that $e^x$ is equal to its Taylor series on the entire real line). In general this can be a hard problem, for instance because the conclusion need not be true! Books on advanced calculus / elementary real analysis will have some worked examples.
More Added: Gosh, for a few minutes there I forgot about the thousand pages or so of lecture notes I have online! In particular see here for some further details on applying Taylor's Formula with Remainder. However this discussion is at a somewhat higher level than strictly necessary (it came from a second semester undergraduate real analysis course). As it happens, starting in January I'll be teaching a sophomore-level course on sequences and series, so in the fullness of time I might have more detailed notes. Anyway, I'm sure that a little googling will find plenty of better notes on this topic...(In fact if you flip towards the back of a good calculus book, you'll certainly find this material there.)
Yet More Added: In response to a direct request, here are the lecture notes generated by my teaching the sophomore level course on sequences and series. The Chapters labelled 0 and 1 were not actually used for the course and are probably wildly inappropriate for most American undergraduate courses on the subject. Chapters 2 and 3 seem (to me, anyway) much more on point, and in fact much of Chapter 3 is a reworking of the older set of noted I linked to above.