Determine the convergence of the series .

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

Problem

Determine the convergence of the series

\displaystyle \sum_{n=1}^{\infty}\frac{\sqrt{n}+2}{n^{\tfrac32}-1}

. (a) It converges, because its terms are asymptotically proportional to

\dfrac1{n^{2}}

. (b) It converges, by the

p

-test with

p=\tfrac32>1

. (c) It diverges, because its terms are asymptotically proportional to

\dfrac1{n}

. (d) It converges, because the terms of the series approach

0

n\to\infty