Determine the nature of the convergence of the series .

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

Problem

Determine the nature of the convergence of the series

\displaystyle \sum_{n=1}^{\infty} (-1)^{n-1}\frac{3n+2}{n^{2}}

. (a) The series converges absolutely since the ratio test yields a limit of

\tfrac34<1

. (b) The series diverges because its terms do not approach

0

. (c) The series converges conditionally; the alternating terms decrease to

0

but the series of absolute values diverges. (d) The series diverges because

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

and

\sum \dfrac{3}{n}

diverges.