Which of the following series converges?
1) $\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n } } { 3 ^ { n } + 2 ^ { n } }$
2) $\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { n + 5 ^ { n } }$
3) $\sum _ { n = 1 } ^ { \infty } \frac { n } { 1 + 4 n }$
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 converges?
1) $\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n } } { 3 ^ { n } + 2 ^ { n } }$ 
2) $\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { n + 5 ^ { n } }$ 
3) $\sum _ { n = 1 } ^ { \infty } \frac { n } { 1 + 4 n }$ 
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Quizplus · Accepted Answer

2) The series $\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { n + 5 ^ { n } }$ converges because the terms in the denominator grow faster than those in the numerator, leading to the terms of the series approaching zero as $n$ approaches infinity. This is a characteristic of convergent series. The other series either grow without bound or do not sufficiently decrease to zero, failing tests for convergence.

Which of the Following Series Converges? 1) ∑n=1∞4n3n+2n\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n } } { 3 ^ { n } + 2 ^ { n } }∑n=1∞​3n+2n4n​

Which of the Following Series Converges?
1) $\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n } } { 3 ^ { n } + 2 ^ { n } }$