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

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

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 $f=g$.