Use the Direct Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { 8 ^ { n } } { 9 ^ { n } + 7 }$.
A)The series $\sum _ { n = 0 } ^ { \infty } \frac { 8 ^ { n } } { 9 ^ { n } + 7 }$ converges.
B)The series $\sum _ { n = 0 } ^ { \infty } \frac { 8 ^ { n } } { 9 ^ { n } + 7 }$ diverges.

Question

Quizplus · Accepted Answer

We can compare the given series with a geometric series with common ratio $\frac{8}{9}$. Specifically, we have for all $n\geq 0$:
$$\frac{8^n}{9^n+7} \leq \frac{8^n}{9^n} = \left(\frac{8}{9}\right)^n$$
Since $\sum_{n=0}^\infty \left(\frac{8}{9}\right)^n$ is a convergent geometric series (with sum $\frac{1}{1-\frac{8}{9}} = 9$), it follows from the Direct Comparison Test that $\sum_{n=0}^\infty \frac{8^n}{9^n+7}$ is also convergent.

Use the Direct Comparison Test to Determine the Convergence or Divergence