We can write many elementary and special functions as power series (for instance, the exponential, trigonometric, Bessel, etc) with infinite radius of convergence, so that for any element $x$ of the domain, even if $x\gg1$, the expression $\sum_{n}c_n x^n$ centered at $x=0$, with some coefficients $c_n$, converges. The questions is then, is there an if and only if for functions to be expressible with $\textit{the same}$ power series for all the elements of the domain ? By "the same" I mean, that the radius of convergence around some unique point is large enough to cover the whole domain.
I think the answer is smoothness but I would like to hear more ideas, or counterexamples if any.
Best Answer
An example contrary to your guess is the function $\frac1{x^2+1}.$ The function is defined at all real numbers, and is infinitely differentiable.
But if you take the power series at $x=a,$ the radius of convergence is $\sqrt{1+a^2}.$
This is because power series, it turns out, are really best studies as complex functions, not real functions.
For example, the above function is defined on all real numbers, but it is undefined on complex numbers $x=\pm i.$ And $\sqrt{1+a^2}$ is the distance from a real $a$ to $\pm i.$
In complex numbers, given an open domain $U\subseteq \mathbb C,$ all functions $g$ which have a single complex derivative have infinite derivatives, and at every point in the domain, the function has a power series of non-zero radius of convergence.
The radius of convergence around a point $a\in U$ is the largest circle for which we can extend the function on the interior of the circle. This can even extend outside our original domain $U.$
For example, the usual function $f(x)=1+x+x^2+\cdots$ is defined in $|x|<1,$ but if you take the power series around $a$ with $|a|<1,$ the radius of convergence will be $|a-1|.$ And these will often extend the definition of $f$ outside the initial domain.
One last thing is that there are infinitely differentiable real functions for which the Taylor series converges, but not to the function. Specifically, there are non-zero functions infinitely differentiable at a point, with all the derivatives zero. The typical example is:
$$f(x)=\begin{cases}e^{-1/x^2}&x\neq 0\\0&x=0\end{cases}$$
Then one can prove that $f^{(n)}(0)=0$ for all $n.$
So the real numbers is the wrong place to be studying power series.