Which of the following series converges?
1) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \sqrt { n } } + \frac { 2 } { n ^ { 3 } } \right)$
2) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { n } - \frac { 1 } { n + 1 } \right)$
3) $\sum _ { n = 1 } ^ { \infty } \frac { \cos ( 1 / n ) } { n ^ { 2 } }$
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 series converges?
1) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \sqrt { n } } + \frac { 2 } { n ^ { 3 } } \right)$ 
2) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { n } - \frac { 1 } { n + 1 } \right)$ 
3) $\sum _ { n = 1 } ^ { \infty } \frac { \cos ( 1 / n ) } { n ^ { 2 } }$ 
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Quizplus · Accepted Answer

2) This series is a telescoping series, where most terms cancel out, leaving a finite sum. 3) This series converges by the comparison test, comparing it to $\sum \frac{1}{n^2}$, which is a p-series with $p > 1$.

Which of the Following Series Converges? 1) ∑n=1∞(1n+2n3)\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \sqrt { n } } + \frac { 2 } { n ^ { 3 } } \right)∑n=1∞​(n​1​+n32​)

Which of the Following Series Converges?
1) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \sqrt { n } } + \frac { 2 } { n ^ { 3 } } \right)$