For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds:
$$\sum_{k=0}^{N} \frac{x^k}{k!} \leq \theta e^{x}$$
In my application, even $N$ is an integer function of $x$, i.e. $N = N(x)$, but for simplicity sake, let's assume $N$ is given for now.
Any ideas?
Thanks for reading
Fred