Calculus Integral Test Calculator

The calculus integral test is a method used to determine whether an infinite series converges or diverges by comparing it to an improper integral. This calculator helps you apply the integral test to your series and interpret the results.

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 useful for positive-term series where the terms are decreasing and continuous.

The test works by comparing the series to an improper integral. If the integral converges, the series also converges. If the integral diverges, the series diverges.

Key Requirements: The function must be continuous, positive, and decreasing for all x ≥ n₀ (some positive integer).

When to Use the Integral Test

The integral test is most effective when:

The series terms can be expressed as a function f(x)
The function is continuous, positive, and decreasing
You need to evaluate the convergence of an infinite series

How to Use This Calculator

To use the integral test calculator:

Enter your function in the input field (e.g., 1/x²)
Specify the lower limit (typically 1 for series starting at n=1)
Click "Calculate" to evaluate the integral
Interpret the result to determine series convergence

Note: The calculator assumes the function is positive and decreasing for x ≥ the lower limit.

Formula Explained

The integral test compares the series Σaₙ to the integral ∫f(x)dx from n₀ to ∞.

If ∫ from n₀ to ∞ f(x) dx converges, then Σ from n=n₀ to ∞ aₙ converges. If ∫ from n₀ to ∞ f(x) dx diverges, then Σ from n=n₀ to ∞ aₙ diverges.

The calculator evaluates the improper integral using numerical methods when an analytical solution isn't possible.

Worked Examples

Example 1: Convergent Series

Consider the series Σ(1/n²) from n=1 to ∞.

The corresponding integral is ∫(1/x²)dx from 1 to ∞, which evaluates to 1. Since the integral converges, the series also converges.

Example 2: Divergent Series

Consider the series Σ(1/n) from n=1 to ∞.

The corresponding integral is ∫(1/x)dx from 1 to ∞, which diverges to infinity. Therefore, the series also diverges.

Frequently Asked Questions

What types of series can the integral test evaluate?

The integral test works best for positive-term series where the terms are continuous, positive, and decreasing.

How accurate are the calculator's results?

The calculator uses numerical integration methods for accuracy. Results are reliable for most practical applications.

Can the integral test determine the exact sum of a series?

No, the integral test only determines convergence or divergence. It doesn't provide the exact sum of the series.