Which of the following statements about the series is true?

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

Problem

Which of the following statements about the series

\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^{p}}

is true? (a) The series converges for all real numbers

p>\tfrac12

and diverges for

p\le\tfrac12

. (b) The series converges only when

p=1

and diverges otherwise. (c) The series converges for all real numbers

p>1

and diverges for

p\le 1

. (d) The series converges for all real numbers

p<1

and diverges for

p\ge 1