For the alternating series U= _ n=1 ^ (-1)^ n+1 1 2^ n
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
For the alternating series
U=n=1∑∞(−1)n+12n1,
consider two different partial sums as candidate approximations of U:
U3=21−41+81 and U5=21−41+81−161+321.
Suppose an approximation is considered acceptable only if its error does not exceed 0.001.
(1) Determine an upper bound for the error when U3 is used to approximate U. Show all work.
(2) Determine an upper bound for the error when U5 is used to approximate U. Show all work.
(3) Indicate which, if any, of the two approximations meet the stated accuracy requirement. Support your conclusion with clear numerical evidence.