SOME BACKGROUND INFO: Analytic functions may be (locally) represented by a convergent power/Taylor series. The domain is given by the interval in which the power series represents this function. For example, $f(x) = e^x$ has $domf = (-∞,∞)$.
In addition, it has also been said that every power series is the Taylor series of some $C^∞$ function.
My question is thus: suppose we had a power series centered at $a$, whose radius of convergence $R = a$. By the above, it must have a Taylor series representation of some analytic, $C^∞$ function. But, also by the above, the domain of this function must be ${a}$, a collapsed interval. How are these ideas compatible? (I feel as if there is some contradiction: a function defined only at one point cannot be differentiated an infinite number of times, and moreover, there would be an infinite number of functions that could be represented by this $R =a $ power series).
Actually, what happens is that every power series with non-zero radius of convergence is the Taylor series of some $C^\infty$ function.
On the other hand, if a Taylor series is centered at $a(>0)$ and if its radius of convergence is $a$, then it converges on the interval $(0,2a)$ and its sum defines a $C^\infty$ function there.