Mathematics

Problem

Let

S=\displaystyle \sum_{n=1}^{\infty} \dfrac{\cos(\pi n)}{\sqrt n}

. Which statement correctly describes the convergence of

S

? a)

S

diverges because

\displaystyle\lim_{n\to\infty}\frac{\cos(\pi n)}{\sqrt n}

oscillates between

1

and

-1

and therefore does not exist. b)

S

converges absolutely since

\dfrac{|\cos(\pi n)|}{\sqrt n}

is a

p

\,-series with

p=\tfrac12>0

, which converges. c)

S

converges conditionally; the alternating series converges while

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

diverges. d)

S

diverges because

\dfrac{1}{\sqrt n}>\dfrac{1}{n}

for all

n

, and the harmonic series

\sum\dfrac{1}{n}

diverges.