Which of the given series are absolutely convergent? Select the correct answer.
A) $\sum _ { n = 1 } ^ { \infty } \frac { \sin 2 n } { n }$
B)
$\sum _ { n = 1 } ^ { \infty } \frac { \cos \frac { \pi n } { 7 } } { n \sqrt { n } }$

Question

Quizplus · Accepted Answer

The series B is absolutely convergent as the terms are bounded above by $\frac{1}{n\sqrt{n}}$ which converges by the p-test. The series A does not converge absolutely as $\sum_{n=1}^{\infty} \frac{|\sin 2n|}{n} \geq \sum_{n=1}^{\infty} \frac{\sin^2 2n}{n} = \frac{1}{2}\sum_{n=1}^{\infty} \frac{1-\cos 4n}{n}$ which is a harmonic series times a bounded sequence, and hence diverges.

Which of the Given Series Are Absolutely Convergent? Select the Correct