Which of the following is true for the infinite series $\sum _ { n = 1 } ^ { \infty } \frac { \cos ( n \pi ) } { n ^ { 2 } } ?$
A)Its sequence of partial sums is not bounded from above.
B)Its sequence of partial sums is decreasing.
C) $\lim _ { n \rightarrow \infty } \frac { \cos ( n \pi ) } { n ^ { 2 } } \neq 0$
D)It is convergent.
E)It is divergent.

Question

Which of the following is true for the infinite series $\sum _ { n = 1 } ^ { \infty } \frac { \cos ( n \pi ) } { n ^ { 2 } } ?$ 
A)Its sequence of partial sums is not bounded from above.
B)Its sequence of partial sums is decreasing.
C) $\lim _ { n \rightarrow \infty } \frac { \cos ( n \pi ) } { n ^ { 2 } } \neq 0$
D)It is convergent.
E)It is divergent.

Quizplus · Accepted Answer

The series $\sum _ { n = 1 } ^ { \infty } \frac { \cos ( n \pi ) } { n ^ { 2 } }$ is convergent because $\cos(n\pi)$ alternates between 1 and -1, making the series an alternating series where the terms $\frac{1}{n^2}$ decrease in absolute value and approach 0, satisfying the conditions for the Alternating Series Test. Additionally, the series can be seen as absolutely convergent because the series of absolute values $\sum _ { n = 1 } ^ { \infty } \frac { |\cos ( n \pi )| } { n ^ { 2 } } = \sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$ is a p-series with $p = 2$, which is known to converge.

Which of the Following Is True for the Infinite Series ∑n=1∞cos⁡(nπ)n2?\sum _ { n = 1 } ^ { \infty } \frac { \cos ( n \pi ) } { n ^ { 2 } } ?∑n=1∞​n2cos(nπ)​?

Which of the Following Is True for the Infinite Series $\sum _ { n = 1 } ^ { \infty } \frac { \cos ( n \pi ) } { n ^ { 2 } } ?$