Which of the following three tests will establish that the series $\sum _ { n = 1 } ^ { \infty } \frac { 3 } { n ( n + 2 ) }$ converges?
1) Comparison Test with $\sum _ { n = 1 } ^ { \infty } 3 n ^ { - 2 }$
2) Limit Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 2 }$
3) Comparison Test with $\sum _ { n = 1 } ^ { \infty } 3 n ^ { - 1 }$
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 $\sum _ { n = 1 } ^ { \infty } \frac { 3 } { n ( n + 2 ) }$ converges?
1) Comparison Test with $\sum _ { n = 1 } ^ { \infty } 3 n ^ { - 2 }$ 
2) Limit Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 2 }$ 
3) Comparison Test with $\sum _ { n = 1 } ^ { \infty } 3 n ^ { - 1 }$ 
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 Comparison Test with $\sum _ { n = 1 } ^ { \infty } 3 n ^ { - 2 }$ works because $\frac { 3 } { n ( n + 2 ) } < 3 n ^ { - 2 }$ for all $n$, and since $\sum _ { n = 1 } ^ { \infty } 3 n ^ { - 2 }$ converges (it's a p-series with $p > 1$), the original series converges.2) The Limit Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 2 }$ also works because the limit of the ratio of the terms of the two series as $n$ approaches infinity is a positive finite number, indicating that both series converge or diverge together. Since $\sum _ { n = 1 } ^ { \infty } n ^ { - 2 }$ converges, so does the original series.3) The Comparison Test with $\sum _ { n = 1 } ^ { \infty } 3 n ^ { - 1 }$ does not work because $\sum _ { n = 1 } ^ { \infty } 3 n ^ { - 1 }$ diverges (it's a harmonic series), and comparing a series to a divergent series in this manner does not establish convergence.

Which of the Following Three Tests Will Establish That the Series