# 9.3 | The Integral Test and p-Series

The Integral Test
If $$f$$ is positive, continuous, and decreasing for $$x \geq 1$$ and $$a_n = f(n)$$, then
$$\sum_{n = 1}^\infty a_n$$
and
$$\int_1^\infty f(x) \; dx$$
either both converge or both diverge.
Convergence of $$p$$-Series
The $$p$$-series
$$\sum_{n=1}^\infty \frac{1}{n^p}$$
converges for $$p >1$$, and diverges for $$p \leq 1$$.