$\text { Use the polynomial test to determine whether the series } \sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 5 } + 9 } \text { converges or }$ diverges.
A) The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 5 } + 9 }$ converges.
B) The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 5 } + 9 }$ diverges.

Question

$\text { Use the polynomial test to determine whether the series } \sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 5 } + 9 } \text { converges or }$ diverges. 
A) The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 5 } + 9 }$ converges.
B) The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 5 } + 9 }$ diverges.

Quizplus · Accepted Answer

We can use the polynomial test to compare the given series to the convergent p-series $\sum_{n=1}^\infty\frac{1}{n^5}$. Note that $n^5+9>n^5$ for all positive integers $n$. Thus, we have $$0<\frac{1}{n^5+9}<\frac{1}{n^5}$$ for all $n\geq1$. Since $\sum_{n=1}^\infty\frac{1}{n^5}$ converges (as it is a p-series with $p>1$), the comparison test implies that $\sum_{n=1}^\infty\frac{1}{n^5+9}$ converges as well. Therefore, the best choice is A.