Integral Test Calculator Wolfram

The Integral Test Calculator helps determine whether an infinite series converges or diverges by comparing it to an improper 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 particularly effective for series where the terms are positive and decreasing.

The test works by comparing the series to an improper integral. If the integral converges, the series may also converge, and if the integral diverges, the series likely diverges.

Key Concepts

Applies to series of positive terms
Requires the function to be continuous, positive, and decreasing
Compares the series to an improper integral

How to Use the Calculator

To use the Integral Test Calculator:

Enter the function f(x) that represents the terms of your series
Specify the lower limit of integration (typically 1 for series)
Click "Calculate" to evaluate the integral
Interpret the result to determine series convergence

For best results, ensure your function is continuous, positive, and decreasing on the interval [a, ∞).

Formula

The Integral Test states that if f(x) is continuous, positive, and decreasing for x ≥ a, then the series Σf(n) from n=1 to ∞ converges if and only if the integral ∫[a,∞] f(x) dx converges.

If ∫[a,∞] f(x) dx converges, then Σf(n) converges.

If ∫[a,∞] f(x) dx diverges, then Σf(n) diverges.

Worked Example

Let's evaluate the series Σ(1/n²) using the Integral Test.

Identify f(x) = 1/x²
Set a = 1
Calculate ∫[1,∞] (1/x²) dx = [ -1/x ] from 1 to ∞ = 1
Since the integral converges to 1, the series Σ(1/n²) converges by the Integral Test

This example shows how the Integral Test can confirm convergence of common series.

FAQ

What types of series can the Integral Test evaluate?: The Integral Test works best for series of positive terms that are decreasing and continuous.
How accurate is the Wolfram integration in this calculator?: The calculator uses Wolfram's numerical integration which provides high accuracy for most functions.
What if the integral doesn't converge or diverge clearly?: In such cases, you may need to use other convergence tests or analyze the function's behavior more carefully.
Can the Integral Test be used for alternating series?: No, the Integral Test is specifically for series of positive terms. Alternating series require different tests.
What if my function doesn't meet all the Integral Test requirements?: You may need to modify your function or use an alternative convergence test if the requirements aren't met.