Which of the following series can be shown to be convergent using the Ratio Test?
1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$
2) $\sum _ { n = 1 } ^ { \infty } \frac { n } { 3 ^ { n } }$
3) $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } } { n ! }$
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 can be shown to be convergent using the Ratio Test?
1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$ 
2) $\sum _ { n = 1 } ^ { \infty } \frac { n } { 3 ^ { n } }$ 
3) $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } } { n ! }$ 
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 effective for series 2 and 3. For series 2, the ratio of successive terms tends to 1/3, and for series 3, the ratio tends to 0 as n increases, both indicating convergence. Series 1, a p-series with p=2, is known to converge, but the Ratio Test is not typically used to demonstrate this.

Which of the Following Series Can Be Shown to Be ∑n=1∞1n2\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }∑n=1∞​n21​

Which of the Following Series Can Be Shown to Be $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$