Consider the series

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

Problem

Consider the series

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

. Using only the

n

th-term test, which conclusion can be drawn? (a) The series converges by the

p

-series test since

p=\tfrac12>1

. (b) The series converges because its terms decrease to zero sufficiently fast. (c) The

n

th-term test is inconclusive because

\displaystyle\lim_{n\to\infty}\frac{1}{\sqrt{n}}=0

. (d) The

n

th-term test cannot be applied here because the terms are not alternating.