Error Bound using Stirling's approximation

Question

Using Stirling's formula, how can we bound the absolute error of $$e^x-\sum_{n=0}^N \frac{x^n}{n!}$$ on the interval $|x|\leq R$ where $R\leq N/2e$

Answer 1

Use the upper bound by geometric series $$ \sum_{n=N+1}^\infty\frac{|x|^n}{n!}\le\frac{|x|^{N+1}}{(N+1)!}\cdot\frac1{1-\frac{|x|}{N+2}} $$ or use the Taylor expansion remainder term $$ e^{\theta x}\cdot\frac{x^{N+1}}{(N+1)!} $$ for some $\theta\in (0,1)$.