Which one of the following series is divergent?
A) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 2 } { 3 } \right) ^ { n }$
B) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } + 1 }$
C) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n 5 ^ { n } }$
D) $\sum _ { n = 2 } ^ { \infty } \frac { n } { n ^ { 2 } - 1 }$
E) $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 3 } } { n ^ { 5 } + 2 }$
F) $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } } { n ! }$
G) $\sum _ { n = 1 } ^ { \infty } \frac { \pi } { n ^ { 2 } }$
H) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { \pi } { n }$

Question

Which one of the following series is divergent?
A) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 2 } { 3 } \right) ^ { n }$ 
B) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } + 1 }$ 
C) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n 5 ^ { n } }$ 
D) $\sum _ { n = 2 } ^ { \infty } \frac { n } { n ^ { 2 } - 1 }$ 
E) $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 3 } } { n ^ { 5 } + 2 }$ 
F) $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } } { n ! }$ 
G) $\sum _ { n = 1 } ^ { \infty } \frac { \pi } { n ^ { 2 } }$ 
H) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { \pi } { n }$

Quizplus · Accepted Answer

Option D is divergent because it simplifies to $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n - 1 } - \frac { 1 } { n + 1 }$, which is a telescoping series that does not converge to a finite limit as $n$ approaches infinity.

Which One of the Following Series Is Divergent? A) ∑n=1∞(23)n\sum _ { n = 1 } ^ { \infty } \left( \frac { 2 } { 3 } \right) ^ { n }∑n=1∞​(32​)n

Which One of the Following Series Is Divergent?
A) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 2 } { 3 } \right) ^ { n }$