Tell which of the following series can be compared with geometric series to establish convergence.1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + 3 ^ { n } }$
2) $\sum _ { n = 1 } ^ { \infty } \frac { n } { n ^ { 3 } + 4 }$
3) $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } } { 3 ^ { n } }$
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Question

Tell which of the following series can be compared with geometric series to establish convergence.1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + 3 ^ { n } }$ 
2) $\sum _ { n = 1 } ^ { \infty } \frac { n } { n ^ { 3 } + 4 }$ 
3) $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } } { 3 ^ { n } }$ 
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) The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + 3 ^ { n } }$ can be compared with a geometric series because the dominant term in the denominator is $3^n$, which makes the series behave similarly to a geometric series with a ratio less than 1, ensuring convergence.2) The series $\sum _ { n = 1 } ^ { \infty } \frac { n } { n ^ { 3 } + 4 }$ cannot be directly compared with a geometric series for convergence because the presence of $n$ in the numerator and $n^3$ in the denominator does not simplify to a form that resembles a geometric series. The behavior of this series is more complex and does not straightforwardly match the criteria for a geometric series.3) The series $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } } { 3 ^ { n } }$ can be compared with a geometric series because both the numerator and the denominator are exponential in $n$, and the denominator grows much faster than the numerator. This makes the series behave similarly to a geometric series with a ratio less than 1, ensuring convergence.

Tell Which of the Following Series Can Be Compared with Geometric