Which of the following series converge?
A) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } + \sin n }$
B) $\sum _ { n = 1 } ^ { \infty } \frac { n } { ( 1 + n ) ^ { 3 } }$
C) $\sum _ { n = 1 } ^ { \infty } \frac { \sin n } { n ^ { 2 } }$
D) all of these
E) none of these

Question

Quizplus · Accepted Answer

Each series provided meets the criteria for convergence. Series A and C converge by the comparison test, comparing them to the p-series $\sum \frac{1}{n^2}$, which converges. Series B converges by the comparison test or the limit comparison test, comparing it to the p-series $\sum \frac{1}{n^2}$, noting that the terms of series B behave similarly to $\frac{1}{n^2}$ for large n.

Which of the Following Series Converge? A) ∑n=1∞1n2+sin⁡n\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } + \sin n }∑n=1∞​n2+sinn1​

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