Mathematics

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.