Consider the series
![\[ \sum_{n=1}^{\infty} \frac{1 - n \sin \frac{1}{n}}{n}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-82ce92e4bb56d74ecf38876dbfb743f4_l3.png "Rendered by QuickLaTeX.com")
Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.
The series converges absolutely.
Proof. We use the comparison test,
![\[ \sum_{n=1}^{\infty} \frac{1-n \sin \frac{1}{n}}{n} \leq \sum_{n=1}^{\infty} \left( 1 - n \sin \frac{1}{n} \right). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-90da8f85e31826b71cc29e4f871258a0_l3.png "Rendered by QuickLaTeX.com")
The series on the right converges absolutely by the previous exercise (Section 10.20, Exercise #31). Hence, by the comparison test, we have established the convergence of the series
![\[ \sum_{n=1}^{\infty} \frac{1 - n \sin \frac{1}{n}}{n}. \qquad \blacksquare\]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-721b2312382f1a23ba041ae72aa564ac_l3.png "Rendered by QuickLaTeX.com")