Let . Which of the following best describes the convergence of…

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

Problem

Let

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

. Which of the following best describes the convergence of

S

? (a)

S

converges absolutely because

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

behaves like

\dfrac{1}{n^{2}}

. (b)

S

converges absolutely by the Ratio Test since the limit of successive terms is less than

1

. (c)

S

converges conditionally because the alternating series converges while the series of absolute values diverges. (d)

S

diverges because the terms are not eventually decreasing, so the Alternating Series Test cannot be applied.