Converges or Diverges Calculator Integral

Determine whether an improper integral converges or diverges using our calculator. This tool helps you apply the comparison test, limit comparison test, ratio test, root test, and integral test to analyze the behavior of integrals at infinity.

What is a Converges or Diverges Integral?

An improper integral is an integral where either the interval of integration is unbounded or the integrand becomes infinite within the interval. These integrals can either converge (have a finite value) or diverge (approach infinity).

An improper integral of the form ∫_a^∞ f(x) dx is said to converge if the limit exists:

lim_b→∞ ∫_a^b f(x) dx exists and is finite.

If the limit does not exist or is infinite, the integral is said to diverge. Common techniques to determine convergence include comparison tests, ratio tests, and integral tests.

How to Use the Calculator

Enter the integrand function f(x) in the input field.
Select the test you want to apply (Comparison Test, Limit Comparison Test, Ratio Test, Root Test, or Integral Test).
Click "Calculate" to determine if the integral converges or diverges.
Review the result and interpretation.

Note: The calculator uses numerical methods to approximate the behavior of the integral. For exact results, symbolic computation software may be required.

Common Convergence Tests

Comparison Test

If 0 ≤ f(x) ≤ g(x) for all x ≥ a, and ∫_a^∞ g(x) dx converges, then ∫_a^∞ f(x) dx also converges.

Limit Comparison Test

If lim_x→∞ [f(x)/g(x)] = L, where 0 < L < ∞, and ∫_a^∞ g(x) dx converges, then ∫_a^∞ f(x) dx also converges.

Ratio Test

For a series ∑a_n, if lim_n→∞ |a_n+1/a_n| = L, then the series converges if L < 1 and diverges if L > 1.

Root Test

For a series ∑a_n, if lim_n→∞ √|a_n| = L, then the series converges if L < 1 and diverges if L > 1.

Integral Test

If f(x) is continuous, positive, and decreasing for x ≥ a, then ∫_a^∞ f(x) dx and ∑_n=a^∞ f(n) either both converge or both diverge.

Worked Examples

Example 1: Using the Comparison Test

Determine if ∫₁^∞ (1/x²) dx converges.

Since 1/x² ≤ 1/x for x ≥ 1, and ∫₁^∞ (1/x) dx diverges, the integral ∫₁^∞ (1/x²) dx also diverges.

Example 2: Using the Integral Test

Determine if ∑_n=1^∞ (1/n²) converges.

Consider the integral ∫₁^∞ (1/x²) dx. Since the integral converges, the series also converges.

Interpreting Results

If the calculator indicates that the integral converges, it means the area under the curve is finite. If it diverges, the area is infinite. Understanding the behavior of the integrand at infinity is crucial for determining convergence.

Test	Condition for Convergence	Condition for Divergence
Comparison Test	∫ g(x) dx converges and f(x) ≤ g(x)	∫ g(x) dx diverges and f(x) ≥ g(x)
Limit Comparison Test	lim [f(x)/g(x)] = L (0 < L < ∞) and ∫ g(x) dx converges	lim [f(x)/g(x)] = ∞ and ∫ g(x) dx diverges
Ratio Test	lim \|a_n+1/a_n\| < 1	lim \|a_n+1/a_n\| > 1
Root Test	lim √\|a_n\| < 1	lim √\|a_n\| > 1
Integral Test	∫ f(x) dx converges	∫ f(x) dx diverges

FAQ

What does it mean for an integral to converge?

An integral converges if the limit of the integral as the upper bound approaches infinity exists and is finite. This means the area under the curve is finite.

How do I know which test to use?

Choose a test based on the form of the integrand. The comparison test works well when you can find a similar function whose convergence is known. The integral test is useful for series where the terms are decreasing and positive.

What if none of the tests apply?

If none of the standard tests apply, you may need to use more advanced techniques or symbolic computation software to determine convergence.