Series Integral Test Calculator with Steps

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 series where the terms are positive and decreasing.

What is the Integral Test?

The Integral Test is a convergence test used 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 will likely diverge.

The Integral Test requires that the function defining the series terms is continuous, positive, and decreasing for all terms beyond some point.

How to Use the Calculator

Enter the function that defines the series terms in the input field.
Specify the lower limit of integration (usually 1 for series starting at n=1).
Click "Calculate" to determine if the series converges or diverges.
Review the step-by-step explanation and visualization of the integral.

Formula Explained

The Integral Test states that if f(n) = aₙ is continuous, positive, and decreasing for all n ≥ N, then:

∫ from N to ∞ f(x) dx converges ⇒ Σ from N to ∞ aₙ converges ∫ from N to ∞ f(x) dx diverges ⇒ Σ from N to ∞ aₙ diverges

Where:

f(x) is the function defining the series terms
N is the starting index of the series
aₙ is the nth term of the series

Worked Example

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

Define f(x) = 1/(x² + 1)
Calculate the integral ∫ from 1 to ∞ of 1/(x² + 1) dx
The integral evaluates to π/2, which is finite
Since the integral converges, the series Σ from n=1 to ∞ of 1/(n² + 1) also converges

Note: The Integral Test provides a sufficient condition for convergence but not a necessary one. Some series may converge even if the integral does not exist.

Limitations

The Integral Test has several limitations:

It only applies to positive, decreasing series
It doesn't provide information about the rate of convergence
It's not a necessary condition for convergence
The integral must exist for the test to be valid

FAQ

When should I use the Integral Test?

Use the Integral Test when dealing with positive, decreasing series where the terms can be expressed as a function of n. It's particularly useful for series that resemble integrals.

What if the integral doesn't exist?

If the integral diverges, the series will likely diverge as well. If the integral converges, the series may converge, but this isn't guaranteed.

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.