Mathematics

Problem

Consider the series

\displaystyle \sum_{n=1}^{\infty}\dfrac{(-1)^n}{\sqrt n}

. Which statement best explains why this series is conditionally convergent but not absolutely convergent? a) Because the ratio test gives a limit of

1

, the series converges but not absolutely. b) The terms decrease to

0

, so the alternating series test guarantees convergence, but

\sum 1/\sqrt n

diverges, so the series is not absolutely convergent. c) Applying the root test to

|a_n|

yields a limit less than

1

, proving absolute convergence. d) Since

\displaystyle\sum 1/\sqrt n

is a

p

-series with

p=\tfrac12>1

, it converges; hence the given series converges absolutely.