The function has the power series expansion f(x)=1-x+x^ 2 -x^ 3 +x^ 4 - , |x|<1
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
The function f(x)=1+x1 has the power series expansion
f(x)=1−x+x2−x3+x4−⋯,∣x∣<1.
Let Pn(x) denote the polynomial consisting of the first n+1 non-zero terms (through xn).
Throughout this problem, let ε=0.001.
(a) Derive an explicit inequality that guarantees ∣f(x)−Pn(x)∣<ε in terms of x and n. Clearly state the term that bounds the error.
(b) Fix n=2. Determine all real x with ∣x∣<1 for which the quadratic approximation P2(x) satisfies the inequality obtained in part (a). Provide your answer as an interval or union of intervals and justify each step in solving the inequality.
(c) For x=0.6, find the smallest integer n such that ∣f(0.6)−Pn(0.6)∣<ε. State both n and the resulting numerical approximation of f(0.6) to six decimal places.