Consider and the series

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

Problem

Consider

d_n = \dfrac{\sqrt[n]{n}-1}{1/n}

and the series

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

. Which of the following correctly describes the behavior predicted by the limit of

d_n

n\to\infty

? (a) The test is inconclusive because

\displaystyle \lim_{n\to\infty} d_n = 0

. (b) The series diverges because

\displaystyle \lim_{n\to\infty} d_n = \infty

. (c) The series diverges because

\displaystyle \lim_{n\to\infty} d_n = 1

. (d) The series converges because

\displaystyle \lim_{n\to\infty} d_n = 0