Classify the series .

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

Problem

Classify the series ∑n=2∞(−1)n+11n ln⁡n\displaystyle \sum_{n=2}^{\infty} (-1)^{n+1} \dfrac{1}{n\,\ln n}. (a) The series converges absolutely. (b) The series diverges because ∑n=2∞1nln⁡n\displaystyle\sum_{n=2}^{\infty}\frac1{n\ln n} diverges. (c) The series converges conditionally but not absolutely. (d) The series converges by the ratio test, so it must converge absolutely.

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