Limit Comparison Test Calculator: How It Works

This calculator applies the limit comparison test to determine whether two infinite series converge or diverge. You enter the general term of your series and a comparison series, and the calculator computes the limit of their ratio, then states the conclusion based on the test’s conditions.

Formula

The limit comparison test states: if you have two series with positive terms, \(\sum a_n\) and \(\sum b_n\), and you compute

\(L = \lim_{n \to \infty} \frac{a_n}{b_n}\),

then:

• If \(0 < L < \infty\), both series converge or both diverge.
• If \(L = 0\) and \(\sum b_n\) converges, then \(\sum a_n\) converges.
• If \(L = \infty\) and \(\sum b_n\) diverges, then \(\sum a_n\) diverges.

Here \(a_n\) and \(b_n\) are the general terms (the nth terms) of the two series. The variable \(n\) is the index, usually a positive integer. The limit is taken as \(n\) approaches infinity.

Worked Example

Using the default inputs: \(a_n = \frac{1}{n^2}\) and \(b_n = \frac{1}{n}\).

1. Compute the ratio: \(\frac{a_n}{b_n} = \frac{1/n^2}{1/n} = \frac{1}{n}\).
2. Take the limit as \(n \to \infty\): \(\lim_{n \to \infty} \frac{1}{n} = 0\).
3. Since \(L = 0\), we look at the comparison series \(\sum \frac{1}{n}\), which is the harmonic series and diverges. The test is inconclusive in this direction (the rule for \(L=0\) requires \(\sum b_n\) to converge).

So the calculator returns \(L = 0\) and notes that the test does not give a conclusion for these defaults. To get a useful result, choose a comparison series whose convergence you already know.

When to Use the Limit Comparison Test

Use this test when your series has a general term that is a fraction of polynomials, roots, or other algebraic expressions, and you can guess a simpler series that behaves similarly for large \(n\). For example, \(\frac{2n^2+3}{5n^4-1}\) behaves like \(\frac{2n^2}{5n^4} = \frac{2}{5n^2}\), so compare with \(\frac{1}{n^2}\).

It is especially useful when the direct comparison test is hard because the inequality is not obvious. The limit comparison test only requires the limit of the ratio, which is often easier to evaluate.

How to Choose a Comparison Series

• Look at the highest power of \(n\) in the numerator and denominator. Drop lower-order terms.
• Simplify the resulting fraction – that is your \(b_n\).
• Check that \(b_n\) is a known convergent or divergent series (like \(p\)-series \(\sum 1/n^p\) or geometric series).
• Make sure both \(a_n\) and \(b_n\) are positive for all sufficiently large \(n\).

Common Mistakes and Misconceptions

One frequent error is using a comparison series that has the same limit behavior but not the same sign – the test requires positive terms. Another is misreading the conclusion when \(L=0\) or \(L=\infty\): these cases only give information if the comparison series converges (for \(L=0\)) or diverges (for \(L=\infty\)). Many students forget that \(L=0\) with a divergent comparison series tells you nothing.

Also, the limit must be evaluated correctly. Use algebraic simplification or L’Hôpital’s rule if needed. A common mistake is plugging in a large number for \(n\) instead of taking the true limit – that only gives an approximation.

Limitations of the Test

The limit comparison test only works for series with positive terms. If your series has negative terms, you may need the alternating series test or absolute convergence. Also, the test does not give the sum of the series – it only tells you whether it converges or diverges. And if the limit \(L\) is 0 or infinity, the test is inconclusive unless you have the right kind of comparison series.

Alternatives

• Direct comparison test: Compare term-by-term if you can establish an inequality.
• Ratio test: Useful for factorials and exponentials.
• Root test: Useful when the general term has an \(n\)th power.
• Integral test: Works when the terms come from a decreasing positive function.

Frequently Asked Questions

What does the calculator output mean?

The calculator gives the limit \(L\). If \(L\) is a positive finite number, your series and the comparison series share the same fate. If \(L=0\) and the comparison series converges, your series converges. If \(L=\infty\) and the comparison series diverges, your series diverges. Otherwise, the test is inconclusive.

Can I use any comparison series?

No. The comparison series must have positive terms and you must know whether it converges or diverges. Typically you choose a \(p\)-series or geometric series because their behavior is well-known.

What if my series has negative terms?

The limit comparison test requires positive terms. If your series has negative terms, consider taking the absolute value (test for absolute convergence) or use a different test like the alternating series test.

Does the limit comparison test give the sum?

No. It only tells you whether the series converges or diverges. Finding the actual sum usually requires other methods, such as recognizing a geometric series or using power series.

Why did my calculator show “inconclusive”?

If \(L=0\) and your comparison series diverges, or \(L=\infty\) and your comparison series converges, the test gives no information. Try a different comparison series that is closer in behavior to your series.

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in a_n | a_n (your series) | - | 1/n^2 | 0
in b_n | b_n (comparison series) | - | 1/n | 0
out L | Limit L = lim(a_n/b_n) | - | a_n / b_n | 4
out result | Conclusion | - | 0 | 0