Mathematics

Problem

Determine the nature of convergence of the series

\displaystyle \sum_{n=1}^{\infty}(-1)^{n+1}\,\dfrac{2n+1}{n^{3/2}}

. a) The series diverges by comparison with

\displaystyle\sum \dfrac1{\sqrt n}

. b) The series converges absolutely since it can be treated as a

p

-series with

p=\tfrac32>1

.},{ c) The series converges conditionally; its terms decrease to

0

, 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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