Which of the three series below converges?
1) $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n \ln n }$
2) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { 2 } }$
3) $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { 3 } }$
A)2
B)1, 2, 3
C)1, 3
D)2, 3
E)3
F)None
G)1, 2
H)1

Question

Which of the three series below converges?
1) $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n \ln n }$ 
2) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { 2 } }$ 
3) $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { 3 } }$ 
A)2
B)1, 2, 3
C)1, 3
D)2, 3
E)3
F)None
G)1, 2
H)1

Quizplus · Accepted Answer

Series 2 and 3 converge by the integral test, as their corresponding functions $ \frac{1}{x(\ln x)^2} $ and $ \frac{1}{x(\ln x)^3} $ are positive, continuous, and decreasing for $ x > 1 $, and their integrals from 1 to infinity converge. Series 1, however, diverges by the integral test, as its corresponding integral diverges.

Which of the Three Series Below Converges? 1) ∑n=2∞1nln⁡n\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n \ln n }∑n=2∞​nlnn1​

Which of the Three Series Below Converges?
1) $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n \ln n }$