1. What is the Root Test?

The Root Test is a convergence test for infinite series that uses the nth root of the absolute value of the terms. For a series ∑aₙ, we calculate L = lim┬(n→∞)√[n]{|aₙ|}. The series converges absolutely if L < 1, diverges if L > 1, and is inconclusive if L = 1.

2. How Does the Calculator Work?

The calculator uses the Root Test formula:

Where:

• \( a_n \) — The nth term of the series
• \( n \) — Term index
• \( L \) — The limit value

Explanation: The calculator approximates the limit by calculating √[n]{|aₙ|} for a large number of terms and observing the trend as n increases.

3. Interpreting the Results

Convergence Criteria:

• If L < 1: The series converges absolutely
• If L > 1: The series diverges
• If L = 1: The test is inconclusive (try another test)

4. Using the Calculator

Tips: Enter the formula for the nth term of your sequence using standard mathematical notation. Use 'n' as the variable. For example: "1/n", "n^2/2^n", or "(-1)^n/n".

5. Frequently Asked Questions (FAQ)

Q1: When should I use the Root Test?
A: The Root Test is particularly useful for series whose terms involve nth powers, such as exponential sequences or sequences with factorial terms.

Q2: What if the limit equals exactly 1?
A: When L = 1, the Root Test is inconclusive. You'll need to use another convergence test such as the Ratio Test, Comparison Test, or Integral Test.

Q3: How many terms should I use for accurate results?
A: Generally, 100-1000 terms provides a good approximation. For slowly converging sequences, you may need more terms.

Q4: Can the calculator handle complex expressions?
A: The calculator supports basic mathematical operations but may have limitations with very complex expressions or special functions.

Q5: What are the limitations of the Root Test?
A: The test may be difficult to apply when the limit is hard to compute analytically, and it's inconclusive when L = 1. Some sequences may also oscillate, making the limit hard to determine numerically.