Use the Comparison Test to determine if the series converges or diverges.
-$$\sum _ { n = 1 } ^ { \infty } \frac { \sin n \cos n } { 4 ^ { n } }$$
A) converges
B) diverges

Question

Quizplus · Accepted Answer

The given series $\sum _ { n = 1 } ^ { \infty } \frac { \sin n \cos n } { 4 ^ { n } }$ can be compared to the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 4 ^ { n } }$, since $|\sin n \cos n| \leq 1$ for all $n$. The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 4 ^ { n } }$ is a geometric series with a ratio less than 1, which converges. Therefore, by the Comparison Test, the original series converges.

Use the Comparison Test to Determine If the Series Converges ∑n=1∞sin⁡ncos⁡n4n\sum _ { n = 1 } ^ { \infty } \frac { \sin n \cos n } { 4 ^ { n } }n=1∑∞​4nsinncosn​

Use the Comparison Test to Determine If the Series Converges $\sum _ { n = 1 } ^ { \infty } \frac { \sin n \cos n } { 4 ^ { n } }$