Use the Direct Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 7 n ^ { 2 } + 9 }$.
A) The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 7 n ^ { 2 } + 9 }$ converges.
B) The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 7 n ^ { 2 } + 9 }$ diverges.

Question

Quizplus · Accepted Answer

We can use the Direct Comparison Test with the convergent p-series to show convergence:
$$\frac{1}{7n^2 + 9} \leq \frac{1}{7n^2} = \frac{1}{7}\cdot\frac{1}{n^2}$$
Since $\sum\frac{1}{n^2}$ is a convergent p-series, by Direct Comparison Test, $\sum\frac{1}{7n^2+9}$ is also convergent.

Use the Direct Comparison Test to Determine the Convergence or Divergence