Series Integral Test Calculator

The Series 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 positive series where the terms are decreasing and continuous.

What is the Integral Test?

The Integral Test is a method to determine the convergence or divergence of an infinite series. It's based on the idea that if the terms of the series are positive, decreasing, and continuous, then the series and its corresponding integral will either both converge or both diverge.

Key requirements for the Integral Test:

The series must consist of positive terms
The terms must be continuous and decreasing
The integral must be evaluated from the first term to infinity

Note

The Integral Test provides a definitive conclusion only when the integral converges or diverges to a finite value. If the integral is indeterminate, the test doesn't provide a clear answer.

How to Use the Calculator

Enter the function that represents the terms of your series in the "Function" field
Specify the lower limit (usually 1 for series starting at n=1)
Click "Calculate" to determine if the series converges or diverges
Review the result and visualization of the integral

The calculator will show you whether the integral converges or diverges, which indicates the behavior of your series.

Formula

Integral Test Formula

Consider the series Σaₙ from n=1 to ∞. If f(x) is continuous, positive, and decreasing for x ≥ 1, then:

If ∫[1,∞) f(x) dx converges, then Σaₙ converges.

If ∫[1,∞) f(x) dx diverges, then Σaₙ diverges.

The calculator applies this test by evaluating the improper integral of your function from the specified lower limit to infinity.

Examples

Example 1: Convergent Series

Consider the series Σ(1/n²) from n=1 to ∞. The corresponding function is f(x) = 1/x².

The integral ∫[1,∞) (1/x²) dx converges to 1, so the series converges.

Example 2: Divergent Series

Consider the series Σ(1/n) from n=1 to ∞. The corresponding function is f(x) = 1/x.

The integral ∫[1,∞) (1/x) dx diverges to infinity, so the series diverges.

Comparison of Series and Their Integrals
Series	Function	Integral	Conclusion
Σ(1/n²)	1/x²	Converges	Series converges
Σ(1/n)	1/x	Diverges	Series diverges
Σ(1/n³)	1/x³	Converges	Series converges

Limitations

The Integral Test has several important limitations:

It only applies to positive series
The terms must be continuous and decreasing
It doesn't provide information about the rate of convergence
If the integral is indeterminate, the test is inconclusive

When to Use Alternatives

For series that don't meet the Integral Test requirements, consider other tests like the Comparison Test, Ratio Test, or Root Test.

FAQ

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

The Integral Test can be applied to positive series where the terms are continuous and decreasing. It's particularly useful for series with terms that can be expressed as a function of n.

What does it mean if the integral converges?

If the corresponding integral converges to a finite value, it suggests that the series also converges. This is because the terms of the series become small enough that their sum approaches a finite limit.

What if the integral diverges?

If the integral diverges to infinity, it indicates that the series also diverges. This is because the terms of the series don't become small enough for their sum to approach a finite limit.

What if the integral is indeterminate?

If the integral is indeterminate (neither clearly convergent nor divergent), the Integral Test doesn't provide a definitive answer. In such cases, you should consider other convergence tests.