Examine the two series below for absolute convergence (A), convergence that is not absolute (C), or divergence (D).
1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n }$
2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } n ^ { - 1 }$
A)1A, 2A
B)1A, 2C
C)1A, 2D
D)1C, 2A
E)1C, 2C
F)1C, 2D
G)1D, 2A
H)1D, 2C

Question

Examine the two series below for absolute convergence (A), convergence that is not absolute (C), or divergence (D).
1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n }$ 
2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } n ^ { - 1 }$ 
A)1A, 2A
B)1A, 2C
C)1A, 2D
D)1C, 2A
E)1C, 2C
F)1C, 2D
G)1D, 2A
H)1D, 2C

Quizplus · Accepted Answer

1) The series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n }$ does not converge because the terms do not approach zero as $n$ approaches infinity. This is a simple alternating series that oscillates between -1 and 1, failing the necessary condition for convergence that the terms of the series must approach zero.2) The series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } n ^ { - 1 }$ is the alternating harmonic series, which converges by the Alternating Series Test. However, it does not converge absolutely because the harmonic series $\sum _ { n = 1 } ^ { \infty } n ^ { - 1 }$ diverges.

Examine the Two Series Below for Absolute Convergence (A), Convergence ∑n=1∞(−1)n\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n }∑n=1∞​(−1)n

Examine the Two Series Below for Absolute Convergence (A), Convergence $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n }$