Use the Ratio Test to examine the two series below, stating: absolute convergence (A), divergence (D), or Ratio Test inconclusive (I).
1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 2 ^ { 2 n + 1 } } { 5 ^ { n } }$
2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 5 ^ { n } } { 2 ^ { 2 n + 1 } }$
A)1A, 2A
B)1A, 2D
C)1A, 2I
D)1D, 2A
E)1D, 2D
F)1D, 2I
G)1I, 2A
H)1I, 2D

Question

Use the Ratio Test to examine the two series below, stating: absolute convergence (A), divergence (D), or Ratio Test inconclusive (I).
1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 2 ^ { 2 n + 1 } } { 5 ^ { n } }$ 
2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 5 ^ { n } } { 2 ^ { 2 n + 1 } }$ 
A)1A, 2A
B)1A, 2D
C)1A, 2I
D)1D, 2A
E)1D, 2D
F)1D, 2I
G)1I, 2A
H)1I, 2D

Quizplus · Accepted Answer

For series 1, the ratio test gives a limit less than 1, indicating absolute convergence. For series 2, the ratio test gives a limit greater than 1, indicating divergence.

Use the Ratio Test to Examine the Two Series Below ∑n=1∞(−1)n−122n+15n\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 2 ^ { 2 n + 1 } } { 5 ^ { n } }∑n=1∞​(−1)n−15n22n+1​

Use the Ratio Test to Examine the Two Series Below $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 2 ^ { 2 n + 1 } } { 5 ^ { n } }$