A convergent alternating series has positive term magnitudes (b_k ) that are strictly…
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
A convergent alternating series has positive term magnitudes bk that are strictly decreasing with k→∞limbk=0. Let RN denote the remainder (error) obtained when the series is truncated after N terms.
(a) Explain, without performing explicit calculations, why ∣RN∣<bN+1∣ for every positive integer N.
(b) Suppose two different partial sums, SN and SM with M>N, are considered. Describe how the inequality in part (a) allows one to justify which of the two partial sums is certain to be at least as accurate as the other.