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^{2}}{\bigl(n^{3}+4\bigr)^{4/3}}

converges or diverges. (a) The series diverges because, for large

n

, its terms behave like

1/n^{1/3}

, and

p<1

. (b) The series converges because, for large

n

, its terms behave like

1/n^{2}

, a

p

-series with

p>1

. (c) The series diverges for the same reason that the harmonic series

\sum 1/n

diverges. (d) The series fails the

n

th-term test for convergence since

\displaystyle\lim_{n\to\infty}\frac{n^{2}}{(n^{3}+4)^{4/3}}\neq 0