Determine the nature of convergence of the series .

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

Problem

Determine the nature of convergence of the series ∑n=1∞(−1)n+1 2n+1n3/2\displaystyle \sum_{n=1}^{\infty}(-1)^{n+1}\,\dfrac{2n+1}{n^{3/2}}. (a) The series diverges by comparison with ∑1n\displaystyle\sum \dfrac1{\sqrt n}. (b) The series converges absolutely since it can be treated as a pp-series with p=32>1p=\tfrac32>1.},{ (c) The series converges conditionally; its terms decrease to 00, but the series of absolute values diverges. (d) The series diverges because the terms are not strictly decreasing, so the alternating-series test fails.

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