Determine whether the series is absolutely convergent

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

Problem

Determine whether the series

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

is absolutely convergent, conditionally convergent, or divergent. (a) Divergent, because

\displaystyle \lim_{n\to\infty} (-1)^{\,n-1}\frac{1}{n}

does not exist. (b) Absolutely convergent, because the

p

-series with

p=1

converges. (c) Conditionally convergent, because the alternating terms decrease to

0

while the series of absolute values diverges. (d) Divergent, because the Ratio Test gives a limit of

1

, which implies divergence.