Integral Test Calculator

Determine the convergence or divergence of an infinite series by comparing it to an improper integral.

Analysis Details

Integral Value:

Condition Check (Positive):

Condition Check (Decreasing):

Visual comparison of the function f(x) and the series terms a(n).

First 10 terms of the series a(n).

n	a(n) Value

What is the Integral Test Calculator?

An integral test calculator is a tool used in calculus to determine whether an infinite series of positive terms converges (sums to a finite value) or diverges (sums to infinity). The test works by comparing the sum of the series’ discrete terms to the area under the curve of a corresponding continuous and decreasing function. This method provides a powerful link between discrete summation and continuous integration.

This calculator is for students, engineers, and mathematicians who need to quickly verify the convergence of a series without performing the manual integration and limit calculations. If the improper integral of the corresponding function converges to a finite value, the series also converges. Conversely, if the integral diverges, so does the series.

The Integral Test Formula and Explanation

The integral test is based on a specific theorem. Suppose you have an infinite series ∑a_n (starting from n=k). Let f(x) be a function such that f(n) = a_n. For the test to be applicable, the function f(x) must satisfy three critical conditions for all x ≥ k:

1. Positive: f(x) > 0
2. Continuous: f(x) has no breaks or jumps.
3. Decreasing: The function’s value gets smaller as x increases.

If these conditions are met, the integral test states:

The series ∑a_n converges if and only if the improper integral ∫_k^∞ f(x) dx is finite (converges).

This means that the series and the integral share the same fate: they either both converge or both diverge. For more information on this, you can check out this article on the integral test. It’s important to note that the value of the integral is NOT equal to the sum of the series.

Variable Explanations

Variable	Meaning	Unit	Typical Range
f(x)	The continuous function corresponding to the series terms an.	Unitless	A mathematical expression of x.
an	The n-th term of the infinite series.	Unitless	A positive, decreasing sequence.
k	The starting index of the series and the lower bound of the integral.	Unitless Integer	k ≥ 1
∫k^∞ f(x) dx	The improper integral from k to infinity.	Unitless	A finite number or infinity.

Practical Examples

Example 1: A Convergent p-series

Let’s analyze the well-known p-series ∑ 1/n², starting from n=1.

• Inputs: Function f(x) = 1/x², Starting term k = 1.
• Conditions: For x ≥ 1, f(x) = 1/x² is positive, continuous, and decreasing.
• Integral: We evaluate ∫₁^∞ (1/x²) dx = [-1/x] from 1 to ∞ = lim_b→∞(-1/b) – (-1/1) = 0 + 1 = 1.
• Result: Since the integral evaluates to a finite number (1), the series converges. This is a classic example of using an integral test calculator.

Example 2: The Divergent Harmonic Series

Consider the harmonic series ∑ 1/n, starting from n=1.

• Inputs: Function f(x) = 1/x, Starting term k = 1.
• Conditions: For x ≥ 1, f(x) = 1/x is positive, continuous, and decreasing.
• Integral: We evaluate ∫₁^∞ (1/x) dx = [ln(x)] from 1 to ∞ = lim_b→∞(ln(b)) – ln(1) = ∞ – 0 = ∞.
• Result: Since the integral diverges to infinity, the series also diverges. A Ratio Test would not be conclusive here, but the integral test is definitive.

How to Use This Integral Test Calculator

1. Enter the Function: In the first input field, type the expression for the n-th term of your series, but use ‘x’ instead of ‘n’. For example, for ∑1/(n²+1), you would enter 1/(x*x + 1). Make sure to use standard JavaScript mathematical functions like Math.pow(x, 3) for x³, Math.log(x) for the natural logarithm, and Math.exp(x) for e^x.
2. Set the Starting Term: In the second field, enter the integer value where your series summation begins (e.g., 1).
3. Calculate: Click the “Calculate Convergence” button.
4. Interpret Results: The calculator will state whether the series converges or diverges. It also provides the approximate value of the integral and an assessment of the test’s conditions.
5. Review Visuals: The chart and table help you visualize the function and the first few terms of the series, confirming the decreasing nature of the terms. A tool like a Root Test calculator could be another way to check your work.

Key Factors That Affect Convergence

• The Power of ‘n’ in the Denominator: For p-series (like ∑1/n^p), convergence is determined entirely by the power ‘p’. If p > 1, the series converges; if p ≤ 1, it diverges.
• Logarithmic Factors: A logarithmic term in the denominator, like in ∑1/(n * ln(n)), can be enough to cause divergence, as the logarithm grows very slowly.
• Exponential Factors: An exponential term in the denominator (e.g., ∑1/eⁿ) causes the terms to shrink very rapidly, almost always guaranteeing convergence.
• The Starting Point ‘k’: The starting term ‘k’ does not affect whether the series converges or diverges. A series is an infinite sum, so a finite number of initial terms won’t change its infinite behavior.
• Function Must Be Decreasing: If the corresponding function f(x) is not eventually decreasing, the integral test cannot be applied.
• Function Must Be Positive: The test is only designed for series with positive terms. For series with negative terms, other tests like the Alternating Series Test are needed. The comparison test is another useful tool in this area.

Frequently Asked Questions (FAQ)

1. What does it mean for a series to converge?

It means that if you add up all the infinite terms of the series, you get a finite, specific number. Divergence means the sum goes to infinity.

2. Can I use the integral test for any series?

No. The test only applies if the series’ terms can be represented by a function that is continuous, positive, and decreasing on the interval of summation.

3. What if the function is not decreasing from the very start?

The test still works if the function is decreasing for all x greater than some number N. The first finite number of terms don’t affect the series’ ultimate convergence or divergence.

4. Is the sum of the series equal to the value of the integral?

No, this is a common misunderstanding. The integral test only tells you if the series converges, not what it converges to. The integral’s value is generally not the same as the series’ sum.

5. How does this calculator handle the “infinity” in the integral?

It uses numerical integration. The calculator computes the integral up to a very large number to approximate its behavior as x approaches infinity. If the value stabilizes, it’s considered convergent.

6. What if I get a “Conditions Not Met” warning?

This means the function you entered is not positive or not decreasing over the test interval. In this case, the integral test is not reliable, and you should use a different convergence test, such as the Ratio Test or comparison test.

7. Why do I need to use ‘x’ instead of ‘n’ in the function?

The calculator’s parser is built on JavaScript, which evaluates functions of ‘x’. The integral test itself is about converting the discrete series (with terms ‘n’) into a continuous function (of a variable ‘x’).

8. What is a p-series?

A p-series is a series of the form ∑1/n^p. The integral test is famously used to prove that p-series converge for p > 1 and diverge for p ≤ 1.