Mathematics

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