Moment generating function $n$ moments

Question

Suppose that a probability distribution $\mu$ on $[0,\infty)$ has only $n$ finite positive moments $\{m_k\}_{k=1}^n$. That is, $m_k = \int_{[0,\infty)}x^k\mu(\text{d} x) < +\infty$ for $k \in \{ 1,\,2,\,\ldots , n\}$ and $m_k = \int_{[0,\infty)}x^k\mu(\text{d} x) = +\infty$ for $k > n$. Is there any justification for approximating the associated moment generating function $\phi(s)$ by the polynomial $$1 + m_1s + m_2\frac{s^2}{2}+ \ldots + m_n\frac{s^n}{n!}$$ for non-positive $s$ near zero?