Chapter 6.4, Problem 242E

Suppose that ${\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{a}_n{x}^n}$converges to a function y such that yⁿ− y' + y = 0 where y(0) = 0 and y(0) = 1. Find a formula that relates a_n+2, a_n+1 and a_nand compute a₁, .... a₅.

The error in approximating the integral ${\displaystyle {\int }_a^{b}f(t)dt}$ by that of a Taylor approximation ${\displaystyle {\int }_a^b{P}_n(t)dt}$is at most ${\displaystyle {\int }_a^b{R}_n(t)dt}$. In the following exercises, the Taylor remainder estimate $R_n\le \frac{M}{(n+1)!}|x-a{|}^{n+1}$ guarantees that the integral of the Taylor polynomial of the given order approximates the integral of f with an error less than $\frac{1}{10}$ .

1. Evaluate the integral of the appropriate Taylor polynomial and verify that it approximates the CAS value with an error less than $\frac{1}{100}$ .

Compare the accuracy of the polynomial integral estimate with the remainder estimate.