Which of the given series are absolutely convergent?
A) $\sum _ { n = 1 } ^ { \infty } \frac { \cos \frac { \pi n } { 8 } } { n \sqrt { n } }$
B) $\sum _ { n = 1 } ^ { \infty } \frac { \sin 5 n } { n }$

Question

Quizplus · Accepted Answer

For absolute convergence, we need to consider the series $\sum _ { n = 1 } ^ { \infty } \frac { | \cos \frac { \pi n } { 8 } | } { n \sqrt { n } }$
As $| \cos \frac { \pi n } { 8 } | \leq 1$, we have $\frac { | \cos \frac { \pi n } { 8 } | } { n \sqrt { n } } \leq \frac { 1 } { n \sqrt { n } }$ for all n. Now, $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n \sqrt { n } }$ is a convergent series using the p-series test with p=3/2. Hence, by comparison test, $\sum _ { n = 1 } ^ { \infty } \frac { | \cos \frac { \pi n } { 8 } | } { n \sqrt { n } }$ is also convergent. Therefore, the given series $\sum _ { n = 1 } ^ { \infty } \frac { \cos \frac { \pi n } { 8 } } { n \sqrt { n } }$ is absolutely convergent.

Regarding series B, we first need to check the convergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { |\sin 5 n| } { n }$. Note that $\sin 5 n$ takes both positive and negative values. Therefore, we consider two cases: 
- Case 1: $\sin 5 n \geq 0$ for infinitely many n.
- Case 2: $\sin 5 n < 0$ for infinitely many n.

In Case 1, we have 
$$\frac { |\sin 5 n| } { n } \leq \frac { \sin 5 n } { n }$$
Using the Dirichlet's test, we have $\sum _ { n = 1 } ^ { \infty } \frac { \sin 5 n } { n }$ is convergent as $ \sum _ { n =1 } ^ { \infty } \frac { 1 } { n } $ is a divergent series and the sequence $\{ \sin 5 n \}$ is a bounded and monotonically decreasing sequence of real numbers. Hence, by comparison test, we conclude that the given series $\sum _ { n = 1 } ^ { \infty } \frac { \sin 5 n } { n }$ is convergent.

In Case 2, we have 
$$\frac { |\sin 5 n| } { n } \leq \frac { | \sin 5 n | } { n } = \frac { \sin (-5 n) } { n } = -\frac { \sin 5 n } { n }$$
Using the same argument as in Case 1, we conclude that the given series $\sum _ { n = 1 } ^ { \infty } \frac { \sin 5 n } { n }$ is convergent.

But this does not tell us anything about the absolute convergence of the given series $\sum _ { n = 1 } ^ { \infty } \frac { \sin 5 n } { n }$, as we cannot apply the comparison test in this case. Therefore, the best choice among the given options is A.

Which of the Given Series Are Absolutely Convergent? A) ∑n=1∞cos⁡πn8nn\sum _ { n = 1 } ^ { \infty } \frac { \cos \frac { \pi n } { 8 } } { n \sqrt { n } }∑n=1∞​nn​cos8πn​​

Which of the Given Series Are Absolutely Convergent?
A) $\sum _ { n = 1 } ^ { \infty } \frac { \cos \frac { \pi n } { 8 } } { n \sqrt { n } }$