Infinite Sequences and Series · Challenge problem

The Maclaurin polynomial $P_n(x)=\displaystyle\sum_{k=0}^{n}\frac{x^k}{k!}$ is used to approximate $e=e^1$. Using the Lagrange error bound together with the estimate $e^c<3$ for $0<c<1$, find the smallest $n$ for which the bound guarantees $|e-P_n(1)|<10^{-4}$.