For which of the following series will the Test for Divergence establish divergence?
1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n }$
2) $\sum _ { n = 1 } ^ { \infty } n ^ { - 1 }$
3) $\sum _ { n = 1 } ^ { \infty } \frac { n + 1 } { 2 n }$
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Question

For which of the following series will the Test for Divergence establish divergence?
1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n }$ 
2) $\sum _ { n = 1 } ^ { \infty } n ^ { - 1 }$ 
3) $\sum _ { n = 1 } ^ { \infty } \frac { n + 1 } { 2 n }$ 
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 Test for Divergence states that if $ \lim _ { n \rightarrow \infty } a _ { n } \neq 0 $, then the series $ \sum _ { n = 1 } ^ { \infty } a _ { n } $ diverges. For series 1, $ \lim _ { n \rightarrow \infty } ( - 1 ) ^ { n } $ does not exist, so the series diverges. For series 3, $ \lim _ { n \rightarrow \infty } \frac { n + 1 } { 2 n } = \frac { 1 } { 2 } \neq 0 $, so the series diverges.

For Which of the Following Series Will the Test for Divergence