Integral Test Calculator with Steps

Determine whether an infinite series converges or diverges using the integral test. This calculator provides step-by-step results and visualizations to help you understand the process.

What is an Integral Test?

The integral test is a method for determining whether an infinite series converges or diverges. It's based on the relationship between series and integrals, particularly the comparison of the series to an integral of a related function.

Key Formula

If \( f(n) = a_n \) is continuous, positive, and decreasing for \( n \geq N \), then the series \( \sum_{n=N}^{\infty} a_n \) and the integral \( \int_{N}^{\infty} f(x) \, dx \) either both converge or both diverge.

When to Use the Integral Test

The integral test is particularly useful when:

The series terms are positive and decreasing
An antiderivative of the series term can be found
Direct comparison tests are difficult to apply

Note: The integral test cannot be applied directly to alternating series or series with negative terms.

How to Use This Calculator

Enter the function \( f(x) \) that represents your series terms, then specify the starting point \( N \) for the integral. The calculator will:

Calculate the improper integral from \( N \) to infinity
Determine convergence or divergence based on the integral's behavior
Display step-by-step results and a visualization

Example Input

Function: \( \frac{1}{x^2} \)
Starting point: 1

Integral Test Methods

There are several approaches to applying the integral test:

Direct Integral Test

Compute the integral directly and analyze its behavior:

If the integral converges, the series converges
If the integral diverges, the series diverges

Limit Comparison Test

Compare the series to a known series using limits:

If \( \lim_{x \to \infty} \frac{a_n}{b_n} = L \) where \( 0 < L < \infty \), then both series either converge or diverge.

Comparison with p-Series

Compare the series to the p-series \( \sum \frac{1}{n^p} \):

If \( p > 1 \), the p-series converges
If \( p \leq 1 \), the p-series diverges

Worked Example

Let's test the series \( \sum_{n=1}^{\infty} \frac{1}{n^3} \):

Step 1: Identify the Function

We have \( a_n = \frac{1}{n^3} \), so \( f(x) = \frac{1}{x^3} \).

Step 2: Compute the Integral

Calculate \( \int_{1}^{\infty} \frac{1}{x^3} \, dx \):

\( \int \frac{1}{x^3} \, dx = -\frac{1}{2x^2} + C \)

Evaluating from 1 to ∞ gives:

\( \lim_{b \to \infty} \left[ -\frac{1}{2b^2} - (-\frac{1}{2}) \right] = \frac{1}{2} \)

Step 3: Analyze the Result

The integral converges to a finite value, so by the integral test, the series \( \sum_{n=1}^{\infty} \frac{1}{n^3} \) also converges.

Comparison of Series and Integral Results
Series	Integral	Conclusion
\( \sum \frac{1}{n^3} \)	\( \int \frac{1}{x^3} \, dx \)	Both converge
\( \sum \frac{1}{n} \)	\( \int \frac{1}{x} \, dx \)	Both diverge

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 an antiderivative exists.
How does the integral test compare to the comparison test?: The integral test is often simpler when the series terms can be expressed as a function of x, while the comparison test requires finding a suitable comparison series.
What if the integral is improper and doesn't converge?: If the integral diverges to infinity, the series also diverges. If the integral converges, the series converges.
Can the integral test be used for alternating series?: No, the integral test is specifically for positive series. Alternating series require different tests like the alternating series test.