Answer
See below.
Work Step by Step
(Theorem 9 section 3)
The Integral test is applicable for the series with positive terms, such that after some index N, the terms decrease.
Then, if we define a continuous real function $f$ such that $f(n)=a_{n}$, for $n\geq N$,
we can test whether $\displaystyle \int_{N}^{\infty}f(x)dx$ converges or not.
The series $\displaystyle \sum_{n=N}^{\infty}a_{n}$ and the integral $\displaystyle \int_{N}^{\infty}f(x)dx$ both converge or both diverge.
The reasoning behind this test is that we test and determine the convergence of a series (which can be very difficult) by testing the convergence of an improper integral (which is relatively simple, or rather, better known to us).
See example 4 of section 3:
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^{2}+1}$ converges because $\displaystyle \int_{1}^{\infty}\frac{1}{x^{2}+1}dx=\frac{\pi}{4}$.
The sum DOES NOT equal the value $\displaystyle \frac{\pi}{4}$, but, because the integral converges, we know that the sum does as well.
