Identify the most appropriate test to be used to determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { n } { 19 n ^ { 2 } + 1 }$ converges or diverges.
A) Limit Comparison Test with $b _ { \mathrm { n } } = \frac { 1 } { n }$
B) Ratio Test
C) Direct Comparison Test with $b _ { \mathrm { n } } = \frac { 1 } { n }$
D) Root Test
E) Alternating Series Test

Question

Quizplus · Accepted Answer

The Limit Comparison Test is most appropriate because the given series $\sum _ { n = 1 } ^ { \infty } \frac { n } { 19 n ^ { 2 } + 1 }$ resembles the form of a p-series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { p } }$ where $p = 1$. By comparing it to $\frac { 1 } { n }$, we can determine convergence or divergence since the behavior of the given series will be similar to that of the comparison series as $n$ approaches infinity.

Identify the Most Appropriate Test to Be Used to Determine ∑n=1∞n19n2+1\sum _ { n = 1 } ^ { \infty } \frac { n } { 19 n ^ { 2 } + 1 }∑n=1∞​19n2+1n​

Identify the Most Appropriate Test to Be Used to Determine $\sum _ { n = 1 } ^ { \infty } \frac { n } { 19 n ^ { 2 } + 1 }$