Which of the three series below converges?
1) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \sqrt [ 5 ] { n ^ { 4 } } } + \frac { 2 } { n ^ { 3 } } \right)$
2) $\sum _ { n = 1 } ^ { \infty } \frac { \ln n } { n ^ { 2 } }$
3) $\sum _ { n = 1 } ^ { \infty } \frac { \sin ( 1 / n ) } { n }$
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 three series below converges?
1) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \sqrt [ 5 ] { n ^ { 4 } } } + \frac { 2 } { n ^ { 3 } } \right)$ 
2) $\sum _ { n = 1 } ^ { \infty } \frac { \ln n } { n ^ { 2 } }$ 
3) $\sum _ { n = 1 } ^ { \infty } \frac { \sin ( 1 / n ) } { n }$ 
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Quizplus · Accepted Answer

1) The first series is a convergent p-series with $p=\frac{5}{4}>1$.2) The second series is a convergent integral test with $f(x)=\frac{\ln x}{x^2}$ which is positive, continuous, decreasing, and $ \int_1^\infty f(x)dx = \lim_{t\to\infty} \int_1^t \frac{\ln x}{x^2} dx = \lim_{t\to\infty} \left[-\frac{1}{x}\ln x + \frac{1}{x}\right]_1^t = \lim_{t\to\infty} \left[-\frac{1}{t}\ln t + \frac{1}{t} + \frac{1}{e}\right] = \frac{1}{e} < \infty $.3) The third series is a divergent series since $ \lim_{n\to\infty} \frac{\sin ( 1 / n ) } { n } \ne 0 $.

Which of the Three Series Below Converges? 1) ∑n=1∞(1n45+2n3)\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \sqrt [ 5 ] { n ^ { 4 } } } + \frac { 2 } { n ^ { 3 } } \right)∑n=1∞​(5n4​1​+n32​)

Which of the Three Series Below Converges?
1) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \sqrt [ 5 ] { n ^ { 4 } } } + \frac { 2 } { n ^ { 3 } } \right)$