Limit of the sum of a factorial series

Question

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)

Answer 1

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.

Answer 2

$$\begin{split} \sum_{i=0}^{n} {\frac{n!}{(n-i)!}} &= n! \sum_{i=0}^{n} {\frac {1}{(n-i)!}}\\ &= n! \sum_{i=0}^{n} {\frac {1}{i!}}\\ &= n!(e +o(1)) \end{split}$$ Therefore you can take $f(n)= e.n!$

Answer 3

For any function $g(n)$ that diverges to infinity,

$$\lim_{n\to\infty}\frac{g(n)+o(g(n))}{g(n)}=1.$$