Which of the following best explains why the harmonic series diverges?

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

Problem

Which of the following best explains why the harmonic series

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

diverges? (a) The series diverges because its terms fail to approach zero as

n\to\infty

. (b) Being a

p

-series with

p>1

, it must diverge by the

p

-series test. (c) Its partial sums grow without bound because the series can be grouped and compared to the divergent integral

\displaystyle \int_{1}^{\infty}\frac{1}{x}\,dx

. (d) Since the ratio

\dfrac{a_{n+1}}{a_n}

equals 1, the ratio test shows the series cannot converge to a finite sum.