Answer
Statements and examples are given.
Work Step by Step
If $f$ is positive, continuous and decreasing for $x\geq1$ and $a_n=f(n)$, then,
$\sum\limits^{\infty}_{n=1}a_nr^n$ and $\int\limits^{\infty}_{1}f(x)dx$
either both converge or both diverge.
Example:
For the series $\sum\limits^{\infty}_{n=1}\frac{n}{n^2+1}$ the function $f(x)=\frac{x}{x^2+1}$ is positive, continuous and decreasing for $x\geq1$
Also $\int\limits^{\infty}_{1}\frac{x}{x^2+1}dx$ diverges, therefore, the series diverges.
