Which of the following series converge?
A) $\sum _ { n = 1 } ^ { \infty } \frac { n } { n ^ { 2 } + 1 }$
B) $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 1 / 2 } } { n ^ { 2 } + 1 }$
C) $\sum _ { n = 1 } ^ { \infty } \frac { n } { n + 1 }$
D) all of these
E) none of these

Question

Quizplus · Accepted Answer

Series B converges because the terms decrease more rapidly than those of a p-series with p > 1, ensuring convergence by comparison to the convergent series $\sum \frac{1}{n^{3/2}}$. Series A and C diverge because their terms do not decrease fast enough to satisfy the conditions for convergence.

Which of the Following Series Converge? A) ∑n=1∞nn2+1\sum _ { n = 1 } ^ { \infty } \frac { n } { n ^ { 2 } + 1 }∑n=1∞​n2+1n​

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