Let , where is a real parameter. Which statement about the convergence of is…

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

Problem

Let

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

, where

p

is a real parameter. Which statement about the convergence of

S(p)

is correct? (a)

S(p)

converges absolutely for

p>1

, converges conditionally for

0<p\le 1

, and diverges for

p\le 0

. (b)

S(p)

converges absolutely for

p\ge 1

, converges conditionally for

0\le p<1

, and diverges for

p<0

. (c)

S(p)

converges absolutely for

p>1

, converges conditionally for

0\le p\le 1

, and diverges for

p<0

. (d)

S(p)

converges absolutely for

p>1

, converges conditionally for

-1<p\le 1

, and diverges for

p\le -1