Introduction
The Integral Test for Convergence states that a positive, continuous, and decreasing series only converges if its improper integral converges.
In other words:
If \sum_{n=1}^{\infty} a_n
is positive, continuous, and decreasing:
\sum_{n=1}^{\infty} a_n
and \int_1^\infty f(x) \, dx
either both converge or both diverge.
Examples
Example 1: p-Series
Consider the series
\sum_{n=1}^{\infty} \frac{1}{n^p}
, where
p > 1
. We can apply the integral test:
\int_1^\infty \frac{1}{x^p} \, dx = \frac{x^{1-p}}{1-p} \Big|_1^\infty
For
p > 1
, the integral converges, thus the series converges.
For
p \leq 1
, the integral diverges, thus the series diverges.
Example 2: Harmonic Series
The harmonic series
\sum_{n=1}^{\infty} \frac{1}{n}
can be tested using the integral:
\int_1^\infty \frac{1}{x} \, dx = \left[ \ln x \right]_1^\infty = \infty
Thus, the harmonic series diverges.