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

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

Problem

Consider the series S=∑n=1∞(−1)n n+2 (−1)nn2.S=\sum_{n=1}^{\infty}(-1)^{n}\,\frac{n+2\,(-1)^{n}}{n^{2}}. The first several terms are −3,  54,  −119,  1916,…-3,\;\tfrac54,\;-\tfrac{11}{9},\;\tfrac{19}{16},\dots. Answer the following questions to determine the behavior of SS. (a) Show that the sequence of absolute term values {n+2(−1)nn2}\Bigl\{\dfrac{n+2(-1)^{n}}{n^{2}}\Bigr\} is not eventually non-increasing. (b) Evaluate lim⁡n→∞n+2(−1)nn2\displaystyle\lim_{n\to\infty}\frac{n+2(-1)^{n}}{n^{2}}. (c) Decide whether SS converges or diverges. Support your answer by citing an appropriate convergence test once the failure in part (a) has been established. (d) Suppose the numerator n+2(−1)nn+2(-1)^{n} is replaced by n+α (−1)nn+\alpha\,(-1)^{n} for a real constant α\alpha. Determine all real values of α\alpha for which the resulting series converges, and justify your conclusion.

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