Let , where is a positive real number. Which statement is…

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

Problem

Let

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

, where

p

is a positive real number. Which statement is correct? (a)

S(p)

converges absolutely for

p>\tfrac{1}{2}

and diverges for

p\le\tfrac{1}{2}

. (b)

S(p)

converges absolutely for

p>1

and diverges otherwise. (c)

S(p)

converges absolutely for

p>1

, converges conditionally for

\tfrac{1}{2}<p\le1

, and diverges for

p\le\tfrac{1}{2}

. (d)

S(p)

converges conditionally for all

p>0

because the numerator is only linear.