*Note* The following strictly speaking does not answer the question but it may answer what the OP *meant*, i.e., under which conditions a formal power series defines a function. 

>    **Definition**  we say that a function $E\left(\lambda\right),$
     analytic in a sectorial region $\Omega=\left\{ z|0<\left|z\right|<B;\left|\textrm{arg}\left(z\right)\right|<\pi/2+\epsilon\right\} ,$
obeys a *strong asymptotic condition* and has $\sum_{n=0}^{\infty}a_{n}\lambda^{n}$
as *strong asymptotic series* (SAS) if there are positive constant $C$ and $\sigma$
such that
$$
\left|E\left(\lambda\right)-\sum_{n=0}^{N}a_{n}\lambda^{n}\right|<C\sigma^{N+1}\left(N+1\right)!\left|\lambda\right|^{N+1}
$$
 for all $N$ and all $\lambda\in\Omega$. 


Given the above one has:

> **Theorem** A SAS defines a function in the sense that if two analitic functions $f,g$ have the same SAS then $f=g$.

*Remark:* Informally the coefficients must not grow too fast. In fact SAS implies  $\left|a_{n}\right|\le Ck^{n}n!$.