Chapter 6.3, Problem 2SP

In this project. we use the Macburin polynomials for e^xto prove that e is irrational. The proof relies on supposing that e is rational and arriving a a contradiction. Therefore, in the following steps, we suppose e = r/s for some integers r and s where s ≠ 0.

2. Let R_n(x) denote the remainder when using p_n(x) to estimate e^x. Therefore, R_n(x) = e^xand R_n(1) = e - p_n(1). Assuming that e= $\frac{r}{s}$ for integers r and s, evaluate R₀(l), R₁(1), R₂(l), R₃(1), R₄(1).