Determine whether the series converges or diverges.

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

Problem

Determine whether the series

\displaystyle\sum_{n=1}^{\infty} n\sin\!\left(\tfrac{1}{n}\right)

converges or diverges. (a) It converges absolutely since for large

n

n\sin\!\left(\frac{1}{n}\right)\le\dfrac{1}{n^{2}}

, and

\sum\dfrac{1}{n^{2}}

converges. (b) It converges because

\displaystyle\lim_{n\to\infty} n\sin\!\left(\frac{1}{n}\right)=0

, so its terms decrease to

0

. (c) It diverges because its terms do not approach

0

. (d) It converges by the Alternating Series Test because

\sin\!\left(\frac{1}{n}\right)

changes sign and decreases in magnitude.