Limit Comparison Test Calculator

An advanced tool to determine the convergence or divergence of infinite series.

Interactive Calculator

Enter the dominant terms of your two series, a_n and b_n, to compare their long-term behavior. For a term like (3n²+5)/(2n³-1), the dominant term is 3n²/2n³.

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Calculation Results

Enter valid series data

Intermediate Values

Limit L = lim_n→∞ (a_n / b_n): N/A

Degree of a_n: N/A

Degree of b_n: N/A

The conclusion about your series ∑a_n will appear here.

What is the Limit Comparison Test?

The limit comparison test calculator is a tool for determining the convergence or divergence of an infinite series. The test itself is a fundamental method in calculus used to analyze an unknown series by comparing it to a series whose convergence properties are already known. For the test to apply, both series, which we can call ∑a_n and ∑b_n, must have positive terms.

This test is particularly useful when the Direct Comparison Test is difficult or inconclusive to apply. Instead of comparing the terms directly (a_n ≤ b_n), the limit comparison test analyzes the limit of their ratio as n approaches infinity.

Limit Comparison Test Formula and Explanation

The core of the test lies in computing a single value, L. The formula is:

L = lim_n→∞ (a_n / b_n)

Here, a_n represents the terms of the series you want to test, and b_n represents the terms of a series you already understand (e.g., a p-series or geometric series). The conclusion depends on the value of L:

1. If 0 < L < ∞ (L is a finite, positive number), then both series share the same fate: they either both converge or both diverge.
2. If L = 0 and the series ∑b_n converges, then the series ∑a_n must also converge.
3. If L = ∞ and the series ∑b_n diverges, then the series ∑a_n must also diverge.

A calculator makes this process simple by computing L and applying these rules automatically. To learn more about how to choose a comparison series, see our guide on the p-series test calculator.

Variables Table

The variables involved in the limit comparison test. All values are unitless mathematical terms.

Variable	Meaning	Unit	Typical Range
an	The general term of the series being tested.	Unitless	Positive real numbers
bn	The general term of the known comparison series.	Unitless	Positive real numbers
L	The limit of the ratio an/bn.	Unitless	0 to ∞
n	The index of the series term.	Integer	1, 2, 3, … to ∞

Practical Examples

Example 1: A Convergent Series

Let’s determine if the series ∑a_n = ∑1/(n² + 3n) converges.

• Inputs: The dominant part of a_n is 1/n². This looks like a p-series. We choose the known convergent series ∑b_n = ∑1/n² (since p=2 > 1).
• Calculation: We compute L = lim_n→∞ [ (1/(n² + 3n)) / (1/n²) ] = lim_n→∞ [ n² / (n² + 3n) ]. Dividing the numerator and denominator by n² gives lim_n→∞ [ 1 / (1 + 3/n) ] = 1.
• Results: Since L = 1 (a finite, positive number) and we know ∑b_n converges, our limit comparison test calculator concludes that ∑a_n also converges.

Example 2: A Divergent Series

Let’s determine if the series ∑a_n = ∑n/(3n² – 100) converges.

• Inputs: The dominant part of a_n is n/3n² = 1/3n. We choose the known divergent harmonic series multiplied by a constant, ∑b_n = ∑1/n (since p=1). For more on harmonic series, see our page on the integral test for convergence.
• Calculation: We compute L = lim_n→∞ [ (n/(3n² – 100)) / (1/n) ] = lim_n→∞ [ n² / (3n² – 100) ]. Dividing by n² gives lim_n→∞ [ 1 / (3 – 100/n²) ] = 1/3.
• Results: Since L = 1/3 (a finite, positive number) and we know ∑b_n diverges, the calculator concludes that ∑a_n also diverges.

How to Use This Limit Comparison Test Calculator

Using the calculator is a straightforward process designed to give you instant, accurate results.

1. Analyze Your Series (a_n): Identify the dominant terms in the numerator and denominator of your series. For example, in (5n³ – n)/(2n⁵ + n²), the dominant terms are 5n³ and 2n⁵.
2. Enter a_n Details: Input the coefficient and power for both the numerator and denominator of a_n into the first four fields.
3. Choose a Comparison Series (b_n): Select a simpler series, typically a p-series (1/n^p), that behaves similarly to a_n. For (5n³ – n)/(2n⁵ + n²), a good choice for b_n would be 1/n², since the overall power is 3-5 = -2.
4. Enter b_n Details: Input the coefficient and power for your chosen b_n. For 1/n², the numerator coefficient is 1, power is 0. The denominator coefficient is 1, power is 2.
5. Set Known Behavior: Use the radio buttons to indicate whether your chosen series ∑b_n converges or diverges. (For ∑1/n², you would select ‘Converges’).
6. Interpret Results: The calculator instantly provides the limit L and a clear conclusion based on the test’s rules. The chart also updates to show the relationship between the series terms. You might find our ratio test calculator helpful for other types of series.

Key Factors That Affect the Limit Comparison Test

• Choice of Comparison Series (b_n): This is the most critical factor. A good choice simplifies the limit calculation. A bad choice might result in a limit of 0 or ∞, which may be inconclusive.
• Dominant Terms: For polynomials or rational functions, only the highest powers of n matter as n → ∞. Lower-order terms become insignificant.
• Positive Terms: The test is only valid for series where a_n > 0 and b_n > 0 for all sufficiently large n. For series with negative terms, you might need an alternating series test.
• Value of L: The conclusion hinges entirely on whether L is finite and positive, zero, or infinite.
• Convergence of ∑b_n: The test is useless if you don’t know whether your comparison series converges or diverges.
• Algebraic Errors: Mistakes in simplifying the fraction a_n/b_n are a common source of incorrect conclusions. Our limit comparison test calculator eliminates this risk.

FAQ

1. When should I use the Limit Comparison Test?

Use it for series with positive terms, especially those that look like rational functions (a polynomial divided by another polynomial). It’s a great alternative when the Direct Comparison Test is hard to set up.

2. How do I choose the comparison series b_n?

Look at the dominant terms of your original series a_n. Simplify the expression by keeping only the highest power of n in the numerator and denominator. The resulting simplified series is usually a good choice for b_n.

3. What if the limit L is 1?

If L=1, it means the terms a_n and b_n behave almost identically for large n. Since 1 is a finite, positive number, the series ∑a_n and ∑b_n either both converge or both diverge.

4. What does it mean if L = 0?

It means the terms of b_n grow significantly faster than a_n. In this case, if the “bigger” series ∑b_n converges, the “smaller” series ∑a_n must also converge. However, if ∑b_n diverges, you learn nothing about ∑a_n.

5. What does it mean if L = ∞?

It means the terms of a_n grow significantly faster than b_n. If the “smaller” series ∑b_n diverges, the “bigger” series ∑a_n must also diverge. If ∑b_n converges, this test is inconclusive.

6. Can I use this test for alternating series?

Not directly. The limit comparison test requires positive terms. However, you can use it to test for the absolute convergence of an alternating series by applying it to the absolute value of the terms.

7. Why are the inputs in this limit comparison test calculator based on coefficients and powers?

Because for limits to infinity of rational functions, only the leading terms matter. This approach simplifies the input process, prevents syntax errors from typing full functions, and directly implements the logic used by experts to solve these problems.

8. Is this test ever inconclusive?

Yes. If L=0 and ∑b_n diverges, the test is inconclusive. Similarly, if L=∞ and ∑b_n converges, the test is inconclusive. In these cases, you must choose a different comparison series b_n or try a different test, like the root test calculator.

Related Tools and Internal Resources

If you found this calculator useful, explore our other calculus tools:

• p-Series Test Calculator: Quickly determine convergence for series in the form 1/n^p.
• Integral Test for Convergence: Use definite integrals to analyze a series’s behavior.
• Ratio Test Calculator: An excellent tool for series involving factorials or n^th powers.
• Root Test Calculator: Another powerful test for series involving n^th powers.
• Alternating Series Test: The specific test for series with alternating signs.
• Absolute vs Conditional Convergence: Understand the different types of convergence for alternating series.