MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Consider the series
S=n=1∑∞n2+1(−1)nn.
Answer the following questions about S.
(1) Determine whether the series formed by the absolute values of the terms of S converges or diverges. Provide complete justification.
(2) Determine whether S itself converges or diverges. Be explicit about any tests or theorems you invoke in your reasoning.
(3) Classify S as absolutely convergent, conditionally convergent, or divergent, and explain how your work in parts (a) and (b) leads to this conclusion.
(4) Suppose the terms of S are rearranged so that every positive term appears first, followed by every negative term. Discuss whether the rearranged series must converge, and if it does converge, whether it must have the same sum as S. Justify your answer using the classification from part (c).