A student claims that the alternating series U= _ n=1 ^ (-1)^ n-1 n 1+n^2 can be…
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
A student claims that the alternating series
U=n=1∑∞(−1)n−11+n2n
can be truncated after the first five terms with an error whose magnitude is less than the magnitude of the sixth term.
(a) Analyze the sequence of term magnitudes en=1+n2n and determine whether it is monotonically decreasing for all n≥1. Provide a rigorous argument or explicit counterexample.
(b) Evaluate n→∞limen and discuss its implication for the student’s claim.
(c) Based on your work in parts (a) and (b), assess the validity of the student’s conclusion. If it is invalid, clearly state why and propose a corrected statement regarding the error bound for partial sums of U.