# Error Bound using Stirling's approximation

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 $R\leq N/2e$

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 $\theta\in (0,1)$.
• Insert the Stirling approximation for $(N+1)!$ and apply the given bounds or similar ones to $\left(\dfrac{|x|e}{N+1}\right)^{N+1}$. – LutzL Jan 17 '18 at 14:11