Ratio and Root Test Calculator

Convergence Tests:

\[ L_{\text{ratio}} = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \] \[ L_{\text{root}} = \lim_{n \to \infty} \sqrt[n]{|a_n|} \]

Test Result:

Conclusion:

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1. What Are Ratio and Root Tests?

The Ratio Test and Root Test are convergence tests for infinite series. They determine whether a series converges absolutely, diverges, or if the test is inconclusive.

2. How the Tests Work

The Ratio Test uses the limit:

\[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]

The Root Test uses the limit:

\[ L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \]

For both tests:

If L < 1, the series converges absolutely
If L > 1, the series diverges
If L = 1, the test is inconclusive

3. Interpreting Results

Details: These tests are particularly useful for series with factorials or exponential terms. The Ratio Test is often easier for series with factorials, while the Root Test may be better for series with nth powers.

4. Using the Calculator

Tips: Enter the general term of your series (aₙ) using standard mathematical notation. For example:

n^2/(2^n)
(n!)/(3^n)
(2n+1)/(n^2+1)

5. Frequently Asked Questions (FAQ)

Q1: Which test should I use?
A: Try the Ratio Test first, especially if the terms involve factorials. Use the Root Test if terms have nth powers.

Q2: What if L = 1?
A: The test is inconclusive. You'll need to try another convergence test like comparison, integral, or alternating series test.

Q3: How many terms should I calculate?
A: For most series, 10-20 terms are sufficient to see convergence behavior.

Q4: Can these tests determine conditional convergence?
A: No, they only test for absolute convergence. For conditional convergence, you need other tests.

Q5: What about series with alternating signs?
A: These tests work on the absolute values of terms, so they're valid for alternating series.