Which of the following series is absolutely convergent?
1) $\sum _ { n = 0 } ^ { \infty } \left( - \frac { 3 } { 4 } \right) ^ { n }$
2) $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n \ln n }$
3) $\sum _ { n = 0 } ^ { \infty } \left( - \frac { 4 } { 3 } \right) ^ { n }$
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 is absolutely convergent?
1) $\sum _ { n = 0 } ^ { \infty } \left( - \frac { 3 } { 4 } \right) ^ { n }$ 
2) $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n \ln n }$ 
3) $\sum _ { n = 0 } ^ { \infty } \left( - \frac { 4 } { 3 } \right) ^ { n }$ 
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Quizplus · Accepted Answer

1) This series is a geometric series with a ratio of $-\frac{3}{4}$, which has an absolute value less than 1, making the series absolutely convergent. 2) The series converges by the alternating series test, but it does not converge absolutely because the harmonic series diverges. 3) This series is a geometric series with a ratio of $-\frac{4}{3}$, which has an absolute value greater than 1, making the series divergent.

Which of the Following Series Is Absolutely Convergent? 1) ∑n=0∞(−34)n\sum _ { n = 0 } ^ { \infty } \left( - \frac { 3 } { 4 } \right) ^ { n }∑n=0∞​(−43​)n

Which of the Following Series Is Absolutely Convergent?
1) $\sum _ { n = 0 } ^ { \infty } \left( - \frac { 3 } { 4 } \right) ^ { n }$