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 \( p \leq 1\).

E 9.3 Exercises