Which of the following series converges?
1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$
2) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } n } { \ln n }$
3) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n }$
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 converges?
1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$ 
2) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } n } { \ln n }$ 
3) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n }$ 
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 option 3, known as the alternating harmonic series, converges by the Alternating Series Test. The series in options 1 and 2 diverge; the first by the Harmonic Series Test, and the second does not meet the conditions for convergence as the terms do not approach 0 fast enough.

Which of the Following Series Converges? 1) ∑n=1∞1n\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }∑n=1∞​n1​

Which of the Following Series Converges?
1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$