Given some $c>0$ fixed, I am interested in upper bounding $$ e^{-c} \sum_{n=N+1}^\infty \frac{1}{n!} c^n \le 1-\frac{\Gamma(N+1,c)}{N!} \le 1-e^{-c} (1+\frac{c}{N+1})^N\le 1-e^{-c} (1+\frac{c}{N+1})^{N+1}, $$ where $\Gamma$ denotes the upper incomplete gamma function, and would like to know the convergence rate for $N\to\infty$.
So here comes my question: Assume $c>0$ fixed. Which lower bounds of the form $$ \left(1+\frac{c}{N}\right)^N \geq l_c(N) \cdot e^c $$ exist for $N\to\infty$? The faster $l_c(N)\to 1$, the better.
As discussed by Nilotpal below, $1-l_c(N)$ can be chosen to be of order $\frac{1}{N}$. But is this rate optimal? Or is there even a tighter way to upper bound the Taylor series from $n=N+1$ of the exponential function?
For $c >0$ and $n \to \infty$, take the first two terms of the Laurent Series expansion about the point $n = \infty$ to get
$$ \left(1+\frac{c}{n}\right)^n > \left(1 - \frac{c^2}{2n}\right)e^c $$
This converges as fast as the $\frac{1}{n}$ term.