use the integral test to determine whether the series shown below converges or diverges. be sure to check…

Explanation:

Step1: Check Integral Test conditions

Let $f(n)=\frac{5n}{n^{2}+9}$. For $n\geq1$, $f(n)>0$. Also, find the derivative $f^\prime(n)=\frac{5(n^{2}+9)-5n(2n)}{(n^{2}+9)^{2}}=\frac{45 - 5n^{2}}{(n^{2}+9)^{2}}$. When $n\geq3$, $f^\prime(n)<0$, so $f(n)$ is eventually decreasing. And $f(n)$ is continuous for $n\geq1$. The conditions of the Integral Test are satisfied.

Step2: Set up the integral

We consider the improper - integral $\int_{1}^{\infty}\frac{5x}{x^{2}+9}dx$. Let $u = x^{2}+9$, then $du=2xdx$. When $x = 1$, $u=10$; as $x\rightarrow\infty$, $u\rightarrow\infty$. The integral becomes $\frac{5}{2}\int_{10}^{\infty}\frac{du}{u}$.

Step3: Evaluate the integral

$\frac{5}{2}\int_{10}^{\infty}\frac{du}{u}=\frac{5}{2}\lim_{t\rightarrow\infty}\int_{10}^{t}\frac{du}{u}=\frac{5}{2}\lim_{t\rightarrow\infty}(\ln u|{10}^{t})=\frac{5}{2}\lim{t\rightarrow\infty}(\ln t-\ln10)=\infty$.

Answer:

B. The series diverges because the conditions of the Integral Test are satisfied and $\int_{1}^{\infty}\frac{5x}{x^{2}+9}dx=\infty$