The Integral Test

Definition

The Integral Test is a method used to determine the convergence or divergence of an infinite series by comparing it to the behavior of a corresponding improper integral. It applies to series $\sum a_n$ where $a_n = f(n)$ for a function $f(x)$ that is continuous, positive, and decreasing.

Why It Matters

In infinite systems, “nearly zero” isn’t enough for stability. The difference between a series that sums to a finite value and one that blows up to infinity is often subtle; the integral test provides a rigorous boundary for convergence by linking discrete sums to continuous areas.

Core Concepts

• The Test: $\sum_{n=1}^{\infty} a_n$ and $\int_{1}^{\infty} f(x) \, dx$ either both converge or both diverge.
  • How to read: “The sum from n equals one to infinity of a n, and the integral from one to infinity of f of x with respect to x, either both converge or both diverge.”
  • Meaning / when to use: When $a_n = f(n)$ with $f$ positive, continuous, and decreasing, the discrete sum and continuous area under the curve grow together—if one is finite, so is the other.

Connected Concepts

• P Series
• The nth-Term Test