Comparison Theorem for Improper Integrals

Definition

The Comparison Theorem for Improper Integrals is a tool used to determine whether an improper integral converges or diverges by comparing it to a known integral. It is particularly useful when the antiderivative of the integrand is difficult or impossible to find.

Why It Matters

It provides a way to determine if an infinite accumulation is finite without needing to find a difficult or non-existent antiderivative.

Core Concepts

Suppose that $f$ and $g$ are continuous functions with $f(x) \geq g(x) \geq 0$ for $x \geq a$.

1. Convergence: If $\int_a^\infty f(x) \, dx$ is convergent, then $\int_a^\infty g(x) \, dx$ is also convergent. (If the “larger” integral is finite, the “smaller” one must also be finite).
   • How to read: “If the integral from a to infinity of f is convergent, then the integral from a to infinity of g is convergent.”
   • Meaning: A smaller non-negative function cannot accumulate more area than a larger one; if the upper bound is finite, so is the lower.
2. Divergence: If $\int_a^\infty g(x) \, dx$ is divergent, then $\int_a^\infty f(x) \, dx$ is also divergent. (If the “smaller” integral is infinite, the “larger” one must also be infinite).
   • How to read: “If the integral from a to infinity of g is divergent, then the integral from a to infinity of f is divergent.”
   • Meaning: If even the smaller function blows up to infinity, the larger one must diverge too.

Connected Concepts

• Improper Integrals
• Comparison Properties of Integrals
• The Integral Test and P Series
• Comparison Tests for Series
• Equivalence
• Order Of Growth
• Trade-offs
• Fundamental Theorem of calculus
• Leverage
• Symmetry
• Taylor Series
• Maclaurin Series
• Scale
• Bottlenecks