Analyze series using ratio and root methods. See convergence verdicts, example tables, formulas, and exports. Build stronger calculus intuition with each tested sequence today.
| n | |an| | |an+1| | |an+1| / |an| | |an|1/n | Interpretation |
|---|---|---|---|---|---|
| 4 | 0.0625 | 0.03125 | 0.5 | 0.5 | Converges absolutely |
| 5 | 0.03125 | 0.015625 | 0.5 | 0.5 | Converges absolutely |
| 6 | 0.015625 | 0.0078125 | 0.5 | 0.5 | Converges absolutely |
This example uses a geometric-style sequence. Both tests give a value below 1, so the series converges absolutely.
Ratio Test: L = lim |an+1 / an|
If L < 1, the series converges absolutely.
If L > 1, the series diverges.
If L = 1, the test is inconclusive.
Root Test: L = lim sup |an|1/n
If L < 1, the series converges absolutely.
If L > 1, the series diverges.
If L = 1, the test is inconclusive.
This calculator can use direct limit entries or finite estimates from your term values.
The ratio test and root test are core tools in calculus. They help you study infinite series with speed and structure. Many series look difficult at first. These tests reduce the problem to a limit. That makes convergence decisions easier. A reliable calculator saves time during homework, revision, and exam practice. It also helps you check each step before writing a formal proof.
This calculator evaluates the ratio test, the root test, or both together. You can enter direct limit values when your algebra is complete. You can also enter numeric term data for a quick estimate. The tool then reports the value used, the source of the value, and the final verdict. This is useful when you want a fast answer and a readable explanation. It is also useful when comparing two methods on the same series.
Use the ratio test for factorials, exponentials, and many power series. Use the root test for nth powers, products, and expressions with exponents tied to n. Both tests work with absolute values. That makes them strong tools for checking absolute convergence. When the value is less than one, the series converges absolutely. When the value is greater than one, the series diverges. When the value equals one, neither test settles the question.
A strong series workflow needs both intuition and accuracy. This page gives you formulas, an example table, export tools, and a clean result block. The result appears above the form, so review is fast. The downloadable report is useful for class notes and saved practice sets. Keep in mind that finite values only estimate the true limit. For a final mathematical argument, always confirm the actual limit behavior as n grows without bound.
The ratio test checks the limit of |an+1/an|. It is often effective for factorial, exponential, and power-series terms.
The root test checks the limit superior of |an|1/n. It is useful when terms contain nth powers or repeated exponent patterns.
If L is less than 1, the series converges absolutely. That means the series of absolute values also converges.
If L is greater than 1, the series diverges. The terms do not shrink fast enough for convergence.
If L equals 1, the test is inconclusive. You need another method, such as comparison, integral, or alternating series analysis.
Enter absolute values for the term fields. Both tests are built around absolute-value expressions, so that is the correct input style.
No. Finite estimates are helpful checks, but the true theorem uses the limit as n becomes very large. Use them as guidance, not the final proof.
The ratio test is very common for power series. The root test can also work well, especially when nth powers appear naturally in coefficients.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.