Chapter 9.2, Problem 74E

Remainders Let

$f(x)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}{x}^k=\frac{1}{1-x}\text{and}{S}_n(x)={\displaystyle \underset{k=0}{\overset{n-1}{\sum }}{x}^k.}}$

The remainder in truncating the power series after n terms is R_n(x) = f(x) − S_n(x), which depends on x.

1. a. Show that R_n(x) = xⁿ/(1 − x).
2. b. Graph the remainder function on the interval |x| < 1 for n = 1, 2, 3. Discuss and interpret the graph. Where on the interval is |R_n(x)| largest? Smallest?
3. c. For fixed n, minimize |R_n(x)| with respect to x. Does the result agree with the observations in part (b)?
4. d. Let N(x) be the number of terms required to reduce |R_n(x)| to less than 10⁻⁶. Graph the function N(x) on the interval |x| < 1. Discuss and interpret the graph.