Determine whether the series converges or diverges.

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

Problem

Determine whether the series

\displaystyle \sum_{n=1}^{\infty} \frac{n}{n^{2}+5}

converges or diverges. (a) It converges because its terms are eventually dominated by

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

. (b) It converges because the series' terms approach

0

n\to\infty

. (c) It diverges because its terms are comparable to

\displaystyle \frac{1}{n}

. (d) It converges because the integral

\displaystyle \int_{1}^{\infty} \frac{n}{n^{2}+5}\,dn

is finite.