The series representation is used to approximate by adding the first three non-zero terms
MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
The series representation ln(1+x)=∑n=1∞(−1)n+1nxn is used to approximate ln(23) by adding the first three non-zero terms. Which of the following statements best explains why the absolute error of this approximation is less than 0.02?
(a) Because x=21 lies within the interval of convergence of the series, any three–term partial sum is automatically within 0.02 of the true value.
(b) Because the error for an alternating series is no greater than the fifth term, and 5(1/2)5=0.00625<0.02, the approximation is within the desired tolerance.
(c) Because the series is alternating with terms that decrease in magnitude to 0, the error is bounded above by the first omitted term, whose value is 0.015625<0.02.
(d) Because the series is geometric with ratio 21, the error after three terms equals 1−211−3=−1, whose magnitude is less than 0.02.