*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. 

**Edit** My memory didn't fail me. The following is taken almost verbatim from Reed and Simon Vol 4 see [here](https://books.google.it/books?id=dnjNCgAAQBAJ&pg=PA43&lpg=PA43&dq=strong+asymptotic+series&source=bl&ots=nyxqxv4tsQ&sig=bKz17Snu06FKc7gqwOOUIWzARrc&hl=en&sa=X&ved=0ahUKEwjnqobmtLTYAhUINxQKHTMBCuEQ6AEIODAC#v=onepage&q=strong%20asymptotic%20series&f=false)

>    **Definition 1**  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  $\left|a_{n}\right|\le C \sigma^{n}n!$. 

There are
series associated with simple examples for which $a_n$ behaves like $(kn)!$ with
$k > 1$. Thus, a strong asymptotic condition cannot hold in such cases. However with a simple modification even this case may be treated. This suggests that we define:


>    **Definition 2**  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 **modified strong asymptotic condition of order $k$** and has $\sum_{n=0}^{\infty}a_{n}\lambda^{n}$
as an **order $k$ strong asymptotic series**  if there are positive constant $C$ and $\sigma$
such that
$$
\left|E\left(\lambda\right)-\sum_{n=0}^{N}a_{n}\lambda^{n}\right|<C\sigma^{N+1}\left [ k(N+1) \right]!\left|\lambda\right|^{N+1}
$$
 for all $N$ and all $\lambda\in\Omega$. 

The above result extends to this case too. Namely

> **Theorem 2** If $f,g$ both have $\sum_{n=0}^{\infty}a_{n}\lambda^{n}$ as order $k$ strong asymptotic series, then $f=g$.