Examine the infinite series _ n=1 ^ (-1)^ n+1 1 n+3 = 14- 15+ 16- 17+ Respond to each…
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Examine the infinite series
n=1∑∞(−1)n+1n+31=41−51+61−71+⋯
Respond to each task below.
(a) Prove that the sequence bn=n+31 is positive, strictly decreasing, and has limit 0 as n→∞.
(b) Decide whether the given series converges. Provide a complete argument based on the properties you established in part (a).
(c) Decide whether the series converges absolutely. Give a rigorous justification that references an appropriate comparison.
(d) Let R20 denote the remainder after the first 20 terms. Establish a numerical upper bound for ∣R20∣ and explain each step of your reasoning.