Examine the two series below for absolute convergence (A), convergence that is not absolute (C), or divergence (D).
1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } n ^ { - 1 }$
2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } n ^ { - 2 }$
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) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } n ^ { - 1 }$ 
2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } n ^ { - 2 }$ 
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) This series is the alternating harmonic series, which converges by the Alternating Series Test, but does not converge absolutely (if you take the absolute value of the terms, you get the harmonic series, which diverges).2) This series converges absolutely because when you take the absolute value of the terms, you get a p-series with p = 2, which converges (p > 1).

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

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