Integral Test Series Calculator

The Integral Test Series Calculator helps determine whether an infinite series converges or diverges by comparing it to an integral. This method is particularly useful for series where the terms are positive and decreasing.

What is the Integral Test?

The Integral Test is a convergence test used in calculus to determine whether an infinite series converges or diverges. It's based on the idea that if the terms of a series are positive and decreasing, the series will converge if the corresponding integral converges.

Key Points

The Integral Test applies to series of the form Σaₙ from n=1 to ∞ where aₙ > 0 and the sequence {aₙ} is decreasing.
If the integral of the function f(x) = aₙ converges, then the series Σaₙ also converges.
If the integral diverges, then the series also diverges.

When to Use the Integral Test

The Integral Test is particularly useful when:

The series terms are positive and decreasing
You can find an antiderivative of the function representing the series terms
You need to determine convergence for a series that doesn't fit other standard tests

How to Use the Calculator

Using the Integral Test Series Calculator is straightforward:

Enter the function representing the series terms in the input field
Specify the lower and upper limits of integration
Click "Calculate" to determine if the series converges or diverges
Review the result and the detailed explanation

Calculator Inputs

Function (f(x)): The function representing the series terms
Lower Limit (a): The lower limit of integration
Upper Limit (b): The upper limit of integration (use ∞ for infinity)

Formula

The Integral Test states that for a series Σaₙ from n=1 to ∞ where aₙ = f(n) and f(x) is continuous, positive, and decreasing for x ≥ 1:

Integral Test Formula

If ∫ from a to ∞ f(x) dx converges, then Σaₙ converges.

If ∫ from a to ∞ f(x) dx diverges, then Σaₙ diverges.

The calculator applies this formula by evaluating the integral of the function you provide.

Example Calculation

Let's examine the series Σ(1/n²) from n=1 to ∞.

Example Steps

Identify f(x) = 1/x²
Calculate the integral ∫ from 1 to ∞ (1/x²) dx
Evaluate the integral: -1/x evaluated from 1 to ∞ = -0 + 1 = 1
Since the integral converges to 1, the series Σ(1/n²) also converges

This example shows how the Integral Test can be applied to determine series convergence.

Frequently Asked Questions

What types of series can the Integral Test be applied to?

The Integral Test is most effective for positive, decreasing series where the terms can be expressed as a function of n. It's particularly useful when other convergence tests are either inconclusive or difficult to apply.

What happens if the integral diverges?

If the corresponding integral diverges, then the series also diverges according to the Integral Test. This provides a clear conclusion when the integral doesn't converge to a finite value.

Can the Integral Test be used for series with negative terms?

The Integral Test is typically applied to series with positive terms. For series with alternating signs or negative terms, other convergence tests like the Alternating Series Test or Absolute Convergence Test would be more appropriate.