Determine whether the following series converges or diverges.
\[
\sum_{n=1}^{\infty} \frac{\cos (n \pi)}{n^{4 / 7}}
\]
Input $C$ for convergence and $D$ for divergence:
Answer
Final Answer: The series \[\sum_{n=1}^{\infty} \frac{\cos (n \pi)}{n^{4 / 7}}\] converges. Therefore, the answer is \(\boxed{C}\).
Step 1 ：Determine whether the following series converges or diverges: \[\sum_{n=1}^{\infty} \frac{\cos (n \pi)}{n^{4 / 7}}\]
Step 2 ：The series is an alternating series, since \(\cos(n\pi)\) alternates between 1 and -1 for integer values of n.
Step 3 ：The terms \(\frac{1}{n^{4/7}}\) are positive, decreasing, and approach 0 as n approaches infinity.
Step 4 ：Therefore, by the Alternating Series Test, the series converges.
Step 5 ：Final Answer: The series \[\sum_{n=1}^{\infty} \frac{\cos (n \pi)}{n^{4 / 7}}\] converges. Therefore, the answer is \(\boxed{C}\).
