Which of the given series are absolutely convergent?
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

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

Quizplus · Accepted Answer

The series in option B is absolutely convergent because it is a p-series with $p = \frac{3}{2}$ (since the denominator is $n^{3/2}$), and for $p > 1$, a p-series converges. The absolute convergence comes from the fact that the absolute value of the cosine function is bounded by 1, making the series comparable to $\sum \frac{1}{n^{3/2}}$, which converges. The series in option A does not guarantee absolute convergence due to the oscillatory nature of the sine function and the denominator being only $n$, making it a harmonic series when considering absolute values, which diverges.

Which of the Given Series Are Absolutely Convergent? A) ∑n=1∞sin⁡2nn\sum _ { n = 1 } ^ { \infty } \frac { \sin 2 n } { n }∑n=1∞​nsin2n​

Which of the Given Series Are Absolutely Convergent?
A) $\sum _ { n = 1 } ^ { \infty } \frac { \sin 2 n } { n }$