Chapter 6.3, Problem 6SP

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.

6. Use Taylor’s theorem to find an estimate on R_n(1). Use this estimate combined with the result from part 5 to show that $|sn!Rn|<\frac{se}{n+1}$ . Conclude that if n is large enough then $|sn!Rn|<1$ . Therefore. sn!R_n(l) is an integer with magnitude less than 1. Thus, .sn!R_n(1) = 0. But from part 5, we know that sn!R_n(l) ? 0. We have arrived at a contradiction, and consequently, the original supposition that e is rational must be false.