Comparison Test for Improper Integrals
Sometimes we can't find the antiderivative (like ), but we still need to know if the integral converges. The Comparison Test lets us determine convergence by comparing to known integrals.
Why Use Comparison?
Many improper integrals have integrands that cannot be integrated using any of our techniques. But we can still determine if the area is finite by comparing the function to one we do know.
The Core Logic
If a larger function converges, the smaller one must also converge (it's trapped underneath).
If a smaller function diverges, the larger one must also diverge (it's even bigger).
Common Comparisons (p-Integrals)
converges if
diverges if
Direct Comparison Test
If for all :
Convergence
If converges,
then also converges.
(Smaller than something finite → finite)
Divergence
If diverges,
then also diverges.
(Bigger than something infinite → infinite)
Limit Comparison Test
When direct comparison is tricky (hard to show which is bigger), use the Limit Comparison Test:
The Formula
If (positive and finite):
→ Both integrals behave the same (both converge or both diverge).
Tip: The Limit Comparison Test is easier because you don't need to prove which function is larger — you just compare their growth rates!
Worked Examples
Example 1: Direct Comparison (Convergent)
Does converge?
Since , we have:
is a p-integral with , so it converges.
Since and the larger integral converges, the smaller integral also converges.
Example 2: Direct Comparison (Divergent)
Does converge?
Since for , we have:
is a p-integral with , so it diverges.
Since and the smaller integral diverges, the larger integral also diverges.
Example 3: Limit Comparison Test
Does converge?
Compare to (p-integral with , converges).
Divide top and bottom by :
Since is finite and positive, both integrals behave the same. Since converges, our integral also converges.
Practice Quiz
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