A student approximates by adding the first four non-zero terms of its Maclaurin series
MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
A student approximates (ln(1.1)) by adding the first four non-zero terms of its Maclaurin series. Which of the following best justifies that the magnitude of the approximation error is less than 0.0003?
(a) Because the terms alternate in sign, their magnitudes decrease to 0, and the magnitude of the first omitted term, 5(0.1)5, is less than 0.0003. (Alternating Series Estimation Theorem)
(b) Each successive term of the series is smaller in magnitude than the previous term, and the fifth term is less than 0.0003. (Alternation not required)
(c) Since ∣x∣=0.1<1, the remainder behaves like a geometric series; the sum of all remaining terms is at most 1−0.1fifth term<0.0003. (Geometric-series bound)
(d) The terms alternate and decrease, so the error is no larger than the fourth term, whose magnitude is 4(0.1)4<0.0003. (Uses last included term)