CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2 · Question 2(a)(iii)

Let F_n(x) = \frac{1}{n!} \int_0^x t^n e^{-t} dt.

Hence, show that if M is an integer greater than 1, then e^x F_M(x) = -\left(x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^M}{M!}\right) + (e^x - 1).