Which of the following three tests will establish that the series $\sum _ { n = 1 } ^ { \infty } \frac { n } { \sqrt { 2 n ^ { 5 } + 1 } }$ converges?
1) Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 5 / 2 }$
2) Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 3 / 2 }$
3) Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 1 / 2 }$
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 three tests will establish that the series $\sum _ { n = 1 } ^ { \infty } \frac { n } { \sqrt { 2 n ^ { 5 } + 1 } }$ converges?
1) Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 5 / 2 }$ 
2) Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 3 / 2 }$ 
3) Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 1 / 2 }$ 
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 Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 3 / 2 }$ is appropriate because the given series $\frac { n } { \sqrt { 2 n ^ { 5 } + 1 } }$ behaves similarly to $n^{-3/2}$ for large $n$, indicating convergence. The other comparisons either diverge or are not suitable comparisons for proving convergence.

Which of the Following Three Tests Will Establish That the Series