Use the Comparison Test to determine if the series converges or diverges.
-$$\sum _ { n = 1 } ^ { \infty } \frac { 5 + 2 \cos n } { n ^ { 4 } }$$
A) diverges
B) converges

Question

Quizplus · Accepted Answer

Note that $|\cos n|\leq 1$ and hence $-1\leq\cos n\leq 1$. Then, we have 
$$\left|\frac{5+2\cos n}{n^4}ight|\leq\frac{5+2}{n^4}=\frac{7}{n^4}$$
Since $\sum_{n=1}^\infty \frac{7}{n^4}$ converges (we can use the $p$-series test with $p > 2$), by the Comparison Test, $\sum_{n=1}^\infty \frac{5+2\cos n}{n^4}$ converges. Thus the correct answer is $\boxed{	ext{(B)}}$.

Use the Comparison Test to Determine If the Series Converges ∑n=1∞5+2cos⁡nn4\sum _ { n = 1 } ^ { \infty } \frac { 5 + 2 \cos n } { n ^ { 4 } }n=1∑∞​n45+2cosn​

Use the Comparison Test to Determine If the Series Converges $\sum _ { n = 1 } ^ { \infty } \frac { 5 + 2 \cos n } { n ^ { 4 } }$