To decide whether the series converges

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

Problem

To decide whether the series

\displaystyle \sum_{n=1}^{\infty}\frac{n^{2}+3}{n^{3}+4\sqrt{n}}

converges, which of the following comparison statements is most appropriate, and what conclusion does it yield? (a) Compare with

\displaystyle \sum \frac{1}{n^{3/2}}

; the given series converges. (b) Compare with

\displaystyle \sum \frac{1}{n^{2}}

; the given series converges. (c) Compare with

\displaystyle \sum \frac{1}{n}

; the given series diverges. (d) Compare with

\displaystyle \sum \frac{1}{n}

; the given series converges.