MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
The natural logarithm function satisfies f(x)=ln(1+x) for −1<x≤1. Engineers wish to approximate ln(1.3) by adding the first n non-zero terms of the Maclaurin series for f. They require the magnitude of the error in the approximation to be less than 10−4.
(a) Write the Maclaurin series for ln(1+x), including a general term, and state its interval of convergence.
(b) Determine the minimum positive integer n such that using the first n non-zero terms to approximate ln(1.3) guarantees the error is less than 10−4. Show the reasoning that leads to your conclusion.
(c) Using that value of n, evaluate the corresponding series approximation to ln(1.3) and give a numerical upper bound for the magnitude of the error.