$For n = 1,2,3,..., let s_{\text{n}} = 1 + \sum_{r=1}^{n} \frac{1}{r!}$
$(a) Prove by mathematical induction that e - s_{\text{n}} = e\int_{0}^{1}\frac{x^n}{n!}e^{ -x}dx
$(b) From (a), deduce that 0 < e - s_{\text{n}} < \frac{3}{(n+1)!}$ $ for n = 1,2,3,.... . [Remember that e < 3 and e^{ -x}...