Let $c > 0$. I'm trying to show that the sequence $\displaystyle\sum\limits_{k=0}^n \left|\frac{n^k-\frac{n!}{(n-k)!}}{n^k}\right|\frac{c^k}{k!}$ converges to zero, as $n \to \infty$.
I know that the nominator is a polynomial of $n$ of degree $k-1$, while the denominator is a polynomial of degree $k$. Therefore the whole expression in the absolute value behaves like $\frac{1}{n}$ for large values of $n$, and the sequence $\sum\limits_{k=0}^n \frac{1}{n}\frac{c^k}{k!}$ obviously converges to zero (it's $\frac{1}{n}$ times the expression for exponent). Therefore my problem is showing that the expression behaves like $\frac{1}{n}$. That's mainly because the polynomial in the nominator's coefficients depend on $k$.
Anybody got an idea?