Formula Used
The ratio test studies an infinite series with terms an.
L = limnāā |an+1 / an|
If L < 1, the series is absolutely convergent.
If L > 1, the series is divergent.
If L = 1, the ratio test is inconclusive.
How to Use This Calculator
- Enter the series term in the an box.
- Use explicit multiplication, such as 3*n.
- Enter a direct ratio expression when you already simplified it.
- Choose sample n values for numerical checking.
- Set a small tolerance for close decisions.
- Press the calculate button.
- Review the estimated limit and decision.
- Download the CSV or PDF report.
Example Data Table
| Series Term |
Ratio Form |
Limit |
Ratio Test Result |
| 1/n! |
1/(n+1) |
0 |
Absolutely convergent |
| n!/n^n |
(n/(n+1))^n |
1/e |
Absolutely convergent |
| n!/3^n |
(n+1)/3 |
Infinity |
Divergent |
| 1/n |
n/(n+1) |
1 |
Inconclusive |
| n^2/3^n |
((n+1)^2/n^2)/3 |
1/3 |
Absolutely convergent |
Understanding the Ratio Test Limit
The ratio test is a direct way to study an infinite series. It compares the size of one term with the next term. The main value is the limit of the absolute ratio. When that limit is below one, the terms shrink fast enough for absolute convergence. When it is above one, the terms do not shrink enough. The series diverges. When the value equals one, the test gives no final answer.
Why This Calculator Helps
Many ratio test examples include factorials, powers, polynomials, or mixed expressions. Manual work can become long. This calculator lets you enter the term expression or the ratio expression directly. Direct ratio mode is useful when the term grows too large for normal decimal arithmetic. The tool checks several n values, estimates the limit, and reports the decision with a tolerance.
Interpreting the Result
A small ratio means each later term is much smaller than the previous term. This supports convergence. A large ratio means the terms stay large or grow. That supports divergence. A ratio close to one needs extra care. You may need a comparison test, p-series test, root test, or alternating series test.
Good Input Practices
Use explicit multiplication for clear expressions. Write 3*n instead of 3n. Use n! for factorials when values are small. For large factorial problems, enter a simplified ratio expression. For example, a term with n! often has a simple next-term ratio after cancellation.
Practical Notes
The displayed sample table is not a proof by itself. It is a numerical guide. A stable pattern gives confidence, but formal work should show algebraic cancellation and the final limit. Use the exported CSV for checking values. Use the PDF option for a clean study record.
Common Series Patterns
Factorial terms often change by one simple factor. Power terms often leave a constant base in the ratio. Polynomial factors usually move toward one after division by the highest power. These patterns explain why the ratio test is popular in calculus and statistics courses. It is especially useful for probability series, likelihood expansions, generating functions, and tail behavior. Always compare the final limit against one, not zero. Record assumptions when a tolerance changes the practical decision.
FAQs
What is the ratio test limit?
It is the limit of the absolute value of an+1 divided by an. The value helps decide whether an infinite series converges absolutely, diverges, or needs another test.
When does the ratio test prove convergence?
The ratio test proves absolute convergence when the limit is less than one. That means each later term becomes smaller fast enough for the series to settle.
When does the ratio test prove divergence?
The test proves divergence when the limit is greater than one. It also shows trouble when ratios grow without bound, because the terms do not shrink properly.
What happens when the limit equals one?
The ratio test becomes inconclusive. The series may converge or diverge. Use another method, such as comparison, integral, p-series, or alternating series testing.
Should I enter the term or the ratio?
Enter the term when values stay manageable. Enter the simplified ratio when the term has factorials, huge powers, or cancellations that make direct evaluation unstable.
Why are sample n values needed?
Sample n values give numerical evidence about the limiting pattern. Larger values often show the trend better, but algebraic proof is still recommended.
Can this calculator handle factorials?
Yes, it supports n! for moderate values. Very large factorials can overflow normal decimal arithmetic, so direct ratio mode is better for those cases.
Is the exported PDF a formal proof?
No. The PDF is a clean report of inputs, samples, and the decision. A formal proof should include algebraic simplification and the exact limit.