Which of the three series below converges?
1) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 2 n - 1 } { 3 n } \right) ^ { n }$
2) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 ^ { n } + n }$
3) $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } + ( - 3 ) ^ { n } } { 3 ^ { n } }$
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 three series below converges?
1) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 2 n - 1 } { 3 n } \right) ^ { n }$ 
2) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 ^ { n } + n }$ 
3) $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } + ( - 3 ) ^ { n } } { 3 ^ { n } }$ 
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) Diverges by the Ratio Test.2) Converges by the Comparison Test with $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 ^ { n } }$.3) Converges by the Limit Comparison Test with $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 3 ^ { n } }$.

Which of the Three Series Below Converges? 1) ∑n=1∞(2n−13n)n\sum _ { n = 1 } ^ { \infty } \left( \frac { 2 n - 1 } { 3 n } \right) ^ { n }∑n=1∞​(3n2n−1​)n

Which of the Three Series Below Converges?
1) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 2 n - 1 } { 3 n } \right) ^ { n }$