Convergence Integral Calculator

Determine whether an improper integral converges or diverges using our precise convergence integral calculator. This tool helps analyze the behavior of integrals at infinity or other singular points, providing clear results and visualizations.

What is Convergence in Integrals?

Convergence in integrals refers to the behavior of an improper integral as it approaches infinity or a singular point. An integral converges if it approaches a finite value, and diverges if it approaches infinity or does not exist.

Improper Integral:

∫_a^∞ f(x) dx = lim_b→∞ ∫_a^b f(x) dx

For an integral to converge, the limit must exist and be finite. Common tests include the p-test, comparison test, and ratio test.

Types of Integral Convergence

There are several types of convergence for improper integrals:

Absolute Convergence: The integral of the absolute value converges.
Conditional Convergence: The integral converges but the absolute value does not.
Divergence: The integral does not approach a finite value.

Note: Absolute convergence implies conditional convergence, but not vice versa.

How to Use This Calculator

Our convergence integral calculator provides a simple interface to test the convergence of improper integrals. Follow these steps:

Enter the integrand function in the input field.
Specify the limits of integration (lower and upper).
Select the type of convergence test to apply.
Click "Calculate" to evaluate the integral.
Review the results and analysis.

The calculator will determine whether the integral converges or diverges and provide additional details about the convergence behavior.

Worked Examples

Let's examine two common examples of convergence testing:

Example 1: Convergent Integral

Consider the integral ∫₁^∞ (1/x²) dx. This integral converges because:

∫ (1/x²) dx = -1/x + C

lim_b→∞ [-1/b + 1] = 1

Example 2: Divergent Integral

Consider the integral ∫₁^∞ (1/x) dx. This integral diverges because:

∫ (1/x) dx = ln|x| + C

lim_b→∞ [ln(b) - ln(1)] = ∞

FAQ

What is the difference between absolute and conditional convergence?

Absolute convergence means the integral of the absolute value of the function converges, while conditional convergence means the integral of the function itself converges but the absolute value does not.

How do I know if an integral converges?

You can use tests like the p-test, comparison test, or ratio test to determine if an integral converges. Our calculator applies these tests automatically.

What happens if an integral diverges?

A divergent integral does not approach a finite value. It may approach infinity or oscillate indefinitely.