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{3n+5}{n^3+2}

. (a) The series diverges because its terms behave like

\dfrac{3}{n}

for large

n

. (b) The series converges because it is a

p

-series with

p=3>1

. (c) The series converges because its terms behave like

\dfrac{3}{n^{2}}

for large

n

. (d) The series converges since

\displaystyle\lim_{n\to\infty}\dfrac{3n+5}{n^{3}+2}=0