Which one of the following series diverge?
A) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ( 2 n + 1 ) }$
B) $\sum _ { n = 1 } ^ { \infty } \frac { 2 n } { n + 1 }$
C) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( n + 1 ) ( n + 3 ) }$
D) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 3 ^ { n } }$
E) $\sum _ { n = 1 } ^ { \infty } \frac { n - 2 } { n 2 ^ { n } }$
F) $\sum _ { n = 1 } ^ { \infty } \frac { 2 n } { n ! }$
G) $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 100 } } { n ! }$
H) $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 100 } } { 2 ^ { n } }$

Question

Which one of the following series diverge?
A) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ( 2 n + 1 ) }$ 
B) $\sum _ { n = 1 } ^ { \infty } \frac { 2 n } { n + 1 }$ 
C) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( n + 1 ) ( n + 3 ) }$ 
D) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 3 ^ { n } }$ 
E) $\sum _ { n = 1 } ^ { \infty } \frac { n - 2 } { n 2 ^ { n } }$ 
F) $\sum _ { n = 1 } ^ { \infty } \frac { 2 n } { n ! }$
G) $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 100 } } { n ! }$ 
H) $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 100 } } { 2 ^ { n } }$

Quizplus · Accepted Answer

Option B diverges because the terms $\frac{2n}{n+1}$ approach 2 as $n$ goes to infinity, not 0. For a series to converge, its terms must approach 0.

Which One of the Following Series Diverge? A) ∑n=1∞1n(2n+1)\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ( 2 n + 1 ) }∑n=1∞​n(2n+1)1​

Which One of the Following Series Diverge?
A) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ( 2 n + 1 ) }$