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Integral Diverge or Converge Calculator

Reviewed by Calculator Editorial Team

Determine whether an improper integral converges or diverges using our calculator. Learn about the different convergence tests, their applications, and how to interpret the results.

What is Integral Convergence?

An improper integral is an integral where either the interval of integration is infinite or the integrand has an infinite discontinuity within the interval. These integrals are called "improper" because they don't fit the standard definition of an integral.

An improper integral is said to converge if the limit of the integral exists and is finite. If the limit does not exist or is infinite, the integral is said to diverge.

Convergence of an integral is different from convergence of a series. While both involve limits, they apply to different mathematical objects.

Tests for Convergence

There are several tests to determine whether an improper integral converges or diverges. The most common ones are:

  1. Direct Comparison Test
  2. Limit Comparison Test
  3. Integral Test
  4. Ratio Test
  5. Root Test

Direct Comparison Test

If f(x) ≥ g(x) ≥ 0 for all x ≥ a, and ∫ from a to ∞ of f(x) dx converges, then ∫ from a to ∞ of g(x) dx also converges.

Limit Comparison Test

If lim (x→∞) [f(x)/g(x)] = L, where 0 < L < ∞, then both ∫ f(x) dx and ∫ g(x) dx either both converge or both diverge.

lim (x→∞) [f(x)/g(x)] = L where 0 < L < ∞

How to Use This Calculator

Our calculator helps you determine whether an improper integral converges or diverges by applying the Limit Comparison Test. Here's how to use it:

  1. Enter the lower limit of integration (a)
  2. Enter the upper limit of integration (b, use ∞ for infinity)
  3. Enter the integrand function f(x)
  4. Enter the comparison function g(x)
  5. Click "Calculate" to see the result

The calculator will compute the limit of f(x)/g(x) as x approaches infinity and determine whether the integral converges or diverges based on the value of L.

Examples

Example 1: Convergent Integral

Consider the integral ∫ from 1 to ∞ of 1/x² dx. Using the comparison function g(x) = 1/x³, we find that lim (x→∞) [1/x² / 1/x³] = ∞. Since L is infinite, we cannot use the Limit Comparison Test directly. Instead, we recognize that ∫ 1/x² dx converges because it's a p-integral with p = 2 > 1.

Example 2: Divergent Integral

Consider the integral ∫ from 1 to ∞ of 1/x dx. Using the comparison function g(x) = 1/x², we find that lim (x→∞) [1/x / 1/x²] = ∞. Again, L is infinite, so we cannot use the Limit Comparison Test directly. However, we know that ∫ 1/x dx diverges because it's a p-integral with p = 1 ≤ 1.

FAQ

What is the difference between convergence and divergence?
An integral converges if the limit of the integral exists and is finite. It diverges if the limit does not exist or is infinite.
Which test should I use to determine convergence?
The choice of test depends on the form of the integrand. Common tests include the Direct Comparison Test, Limit Comparison Test, and Integral Test.
What does it mean if the limit L is infinite?
If the limit L is infinite, the Limit Comparison Test cannot be applied directly. You may need to use another test or recognize the integral as a standard form.
Can an integral converge but not have a finite value?
No, by definition, an integral must have a finite value to be considered convergent.