Which of the following is true for the infinite series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( n + 4 ) } { ( 7 n + 5 ) ^ { 3 } } ?$
A)Its sequence of partial sums is not bounded from above.
B)Its sequence of partial sums is decreasing.
C)It is absolutely convergent.
D)It is conditionally convergent.
E)It is divergent.

Question

Which of the following is true for the infinite series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( n + 4 ) } { ( 7 n + 5 ) ^ { 3 } } ?$ 
A)Its sequence of partial sums is not bounded from above.
B)Its sequence of partial sums is decreasing.
C)It is absolutely convergent.
D)It is conditionally convergent.
E)It is divergent.

Quizplus · Accepted Answer

The series is absolutely convergent because the series formed by taking the absolute values of its terms, $\sum _ { n = 1 } ^ { \infty } \frac { | ( - 1 ) ^ { n } ( n + 4 ) | } { ( 7 n + 5 ) ^ { 3 } } = \sum _ { n = 1 } ^ { \infty } \frac { n + 4 } { ( 7 n + 5 ) ^ { 3 } }$, converges by the comparison test with a $p$-series, where $p > 1$.

Which of the Following Is True for the Infinite Series ∑n=1∞(−1)n(n+4)(7n+5)3?\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( n + 4 ) } { ( 7 n + 5 ) ^ { 3 } } ?∑n=1∞​(7n+5)3(−1)n(n+4)​?

Which of the Following Is True for the Infinite Series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( n + 4 ) } { ( 7 n + 5 ) ^ { 3 } } ?$