Consider the alternating series S= _ n=1 ^ (-1)^ n+1 1 3n

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider the alternating series S=∑n=1∞(−1)n+113n.\textstyle S=\sum_{n=1}^{\infty}(-1)^{n+1}\frac1{3n}. The remainder RN=S−SNR_N=S-S_N is the difference between the infinite sum SS and the NNth partial sum SNS_N. (a) State the two conditions on the positive term sequence {an}\{a_n\} that ensure the magnitude of the remainder satisfies ∣RN∣≤aN+1|R_N|\le a_{N+1} when a finite partial sum is used to approximate an alternating series whose terms are (−1)n+1an(-1)^{n+1}a_n. Provide clear verbal descriptions rather than merely symbols. (b) Verify that the sequence an=13n\displaystyle a_n=\frac1{3n} satisfies each condition you listed in part (a). Be explicit in your reasoning, citing appropriate limits or inequalities. (c) Explain, in complete sentences, why the verification in part (b) allows one to estimate the error in using SNS_N to approximate SS simply by examining the first omitted term.

Related theory

Browse the problem library