Limit of the sum of a factorial series Is there any function $f$ over the positive integers such that

$$\lim_{n\rightarrow\infty}\frac{\sum_{i=0}^{n} {\frac{n!}{(n-i)!}}}{f(n)} = 1$$

and

$$f(n)\not\equiv\sum_{i=0}^{n} {\frac{n!}{(n-i)!}}$$
What is the definition of the function if it exists, or if it doesn't exist, why not?

(updated)
$$\begin{split}
\sum_{i=0}^{n} {\frac{n!}{(n-i)!}} &amp;= n! \sum_{i=0}^{n} {\frac {1}{(n-i)!}}\\
&amp;= n! \sum_{i=0}^{n} {\frac {1}{i!}}\\
&amp;= n!(e +o(1))
\end{split}$$
Therefore you can take $f(n)= e.n!$
As it stands the question has an obvious answer: take $f(n) = \sum\limits_{i=0}^{n} \frac {n!} {(n-i)!}+1$. 

PS If $a_n \to \infty$ there are lots of sequences $f(n)$ such that $\frac {a_n} {f(n)} \to 1$; for example $f(n)=a_n+c\sqrt {a_n}$ (where $c$ is a constant) is one such.