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 $\left|a_{n}\right|\le Ck^{n}n!$.