Which of the following series converges?
1) $\sum _ { n = 1 } ^ { \infty } \sin \left( \frac { 1 } { n ^ { 2 } } \right)$
2) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt [ 3 ] { n } }$
3) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 3 n + 1 } { 2 n + 1 } \right) ^ { n }$
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 converges?
1) $\sum _ { n = 1 } ^ { \infty } \sin \left( \frac { 1 } { n ^ { 2 } } \right)$ 
2) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt [ 3 ] { n } }$ 
3) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 3 n + 1 } { 2 n + 1 } \right) ^ { n }$ 
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 $\sum _ { n = 1 } ^ { \infty } \sin \left( \frac { 1 } { n ^ { 2 } } \right)$ converges by the comparison test, since $\sin \left( \frac { 1 } { n ^ { 2 } } \right) \leq \frac { 1 } { n ^ { 2 } }$ for all $n$, and the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$ is a convergent p-series with $p = 2 > 1$.2) The series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt [ 3 ] { n } }$ converges by the alternating series test, since $\frac { 1 } { \sqrt [ 3 ] { n } }$ is decreasing to 0.3) The series $\sum _ { n = 1 } ^ { \infty } \left( \frac { 3 n + 1 } { 2 n + 1 } \right) ^ { n }$ diverges because the terms do not go to 0 as $n$ goes to infinity.

Which of the Following Series Converges? 1) ∑n=1∞sin⁡(1n2)\sum _ { n = 1 } ^ { \infty } \sin \left( \frac { 1 } { n ^ { 2 } } \right)∑n=1∞​sin(n21​)

Which of the Following Series Converges?
1) $\sum _ { n = 1 } ^ { \infty } \sin \left( \frac { 1 } { n ^ { 2 } } \right)$