Use the Limit Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 9 n } { 9 n ^ { 2 } + 2 }$.
A) The series $\sum _ { n = 1 } ^ { \infty } \frac { 9 n } { 9 n ^ { 2 } + 2 }$ converges.
B) The series $\sum _ { n = 1 } ^ { \infty } \frac { 9 n } { 9 n ^ { 2 } + 2 }$ diverges.

Question

Quizplus · Accepted Answer

The Limit Comparison Test with the series $\sum \frac{1}{n}$ shows that the given series diverges because the limit of the ratio of $\frac { 9 n } { 9 n ^ { 2 } + 2 }$ to $\frac{1}{n}$ as $n$ approaches infinity is a positive constant, indicating similar behavior to the harmonic series, which is known to diverge.

Use the Limit Comparison Test to Determine the Convergence or Divergence