Tell which of the following three series cannot be found convergent by the Ratio Test but can be found convergent by comparison with a p-series.
1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + 3 ^ { n } }$
2) $\sum _ { n = 1 } ^ { \infty } \frac { n } { n ^ { 3 } + 4 }$
3) $\sum _ { n = 1 } ^ { \infty } \frac { \sqrt { n } } { n ^ { 2 } + n }$
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 three series cannot be found convergent by the Ratio Test but can be found convergent by comparison with a p-series.
1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + 3 ^ { n } }$ 
2) $\sum _ { n = 1 } ^ { \infty } \frac { n } { n ^ { 3 } + 4 }$ 
3) $\sum _ { n = 1 } ^ { \infty } \frac { \sqrt { n } } { n ^ { 2 } + n }$ 
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Quizplus · Accepted Answer

The Ratio Test is inconclusive for series that behave like p-series with p > 1. Series 2 and 3 can be compared to p-series with p > 1 and shown to converge, but the Ratio Test does not provide a conclusive result for them.

Tell Which of the Following Three Series Cannot Be Found ∑n=1∞12+3n\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + 3 ^ { n } }∑n=1∞​2+3n1​

Tell Which of the Following Three Series Cannot Be Found $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + 3 ^ { n } }$