Integral test

Definition

The integral test is a method to determine the convergence or divergence of an infinite series by comparing it to an improper integral. If the integral converges, so does the series, and if the integral diverges, so does the series.

5 Must Know Facts For Your Next Test

1. The function must be continuous, positive, and decreasing on $[a, \infty)$ for the integral test to apply.
2. If $\int_a^\infty f(x) \, dx$ converges, then $\sum_{n=a}^{\infty} f(n)$ also converges.
3. If $\int_a^\infty f(x) \, dx$ diverges, then $\sum_{n=a}^{\infty} f(n)$ also diverges.
4. The integral test helps establish a direct connection between improper integrals and infinite series.
5. When using this test, selecting appropriate bounds for integration is crucial to correctly determining convergence or divergence.

Review Questions

• What conditions must a function meet for the integral test to be applicable?
• How do you determine if an infinite series converges or diverges using the integral test?
• What is the significance of comparing an infinite series to an improper integral in determining convergence?

Related terms

Convergence:

A property where the sum of all terms in a sequence approaches a finite limit as more terms are added.

Divergence:

A property where the sum of all terms in a sequence increases without bound as more terms are added.

$p$-Series Test: $p$-series are of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$; they converge if $p > 1$ and diverge if $0 < p \leq 1$.