Which of the following series are convergent, but not absolutely convergent?
1) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt [ 3 ] { n } }$
2) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { \sqrt { n } \ln n }$
3) $\sum _ { n = 1 } ^ { \infty } \frac { \cos n } { 2 ^ { n } }$
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Question

Which of the following series are convergent, but not absolutely convergent?
1) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt [ 3 ] { n } }$ 
2) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { \sqrt { n } \ln n }$ 
3) $\sum _ { n = 1 } ^ { \infty } \frac { \cos n } { 2 ^ { n } }$ 
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Quizplus · Accepted Answer

The series in options 1 and 2 are conditionally convergent. They converge by the alternating series test but do not converge absolutely because their corresponding series of absolute values diverge.

Which of the Following Series Are Convergent, but Not Absolutely ∑n=1∞(−1)nn3\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt [ 3 ] { n } }∑n=1∞​3n​(−1)n​

Which of the Following Series Are Convergent, but Not Absolutely $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt [ 3 ] { n } }$