Classify the convergence of the series .

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Classify the convergence of the series ∑n=1∞(−1)n+1 3n−1(n2+2)\displaystyle \sum_{n=1}^{\infty}(-1)^{n+1}\,\dfrac{3n-1}{(n^{2}+2)}. (a) Diverges because its terms behave like 1n\dfrac{1}{n}, and the alternating factor cannot change this. (b) Converges absolutely because its terms resemble 1n2\dfrac{1}{n^{2}}. (c) Converges conditionally; the alternating series converges, but the series of absolute values behaves like 1n\dfrac{1}{n}. (d) Converges absolutely by the Ratio Test since lim⁡n→∞∣an+1an∣<1\displaystyle \lim_{n\to\infty}\bigl|\tfrac{a_{n+1}}{a_{n}}\bigr|<1.

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