Absolute Ratio Test Calculator
Determine the convergence of infinite series using the Ratio Test with step-by-step analysis
Series Information
Use standard mathematical notation (n for index, ^ for exponents)
Ratio Test Results
Enter your series to see detailed convergence analysis.
Step-by-Step Solution
Your detailed solution steps will appear here.
📚 Common Series Examples
| Series Type | General Term | Ratio Test Limit (L) | Convergence |
|---|---|---|---|
| Geometric | rⁿ | |r| | Converges if |r| < 1 |
| Exponential | 1/n! | 0 | Always converges |
| p-Series | 1/nᵖ | 1 | Inconclusive (use p-test) |
| Alternating | (-1)ⁿ/n | 1 | Converges (Leibniz) |
| Factorial | n!/10ⁿ | ∞ | Diverges |
| Power Series | xⁿ/n! | 0 | Converges ∀x |
Note: The Ratio Test is inconclusive when L = 1 (use other tests in these cases).
📚 Ratio Test Tips & Theory
Ratio Test Theorem
For a series Σaₙ, compute L = lim|aₙ₊₁/aₙ|. If L < 1: converges absolutely; L > 1: diverges; L = 1: inconclusive.
When to Use
Best for series with factorials, exponentials, or terms with n in exponents. Less effective for p-series.
Common Mistakes
1. Forgetting absolute values. 2. Incorrect limit calculation. 3. Misapplying when L=1. 4. Wrong term ratio.
Related Tests
Root Test (similar), Comparison Test (for p-series), Integral Test (for decreasing terms), Alternating Series Test.
Limit Calculation
Simplify aₙ₊₁/aₙ before taking limit. Use L'Hôpital's Rule for indeterminate forms like ∞/∞ or 0/0.
Absolute Convergence
If Σ|aₙ| converges, then Σaₙ converges absolutely (and thus converges). Stronger than regular convergence.
Note: This calculator provides an automated analysis using the Ratio Test. For series where L=1, additional convergence tests may be needed. Always verify results with manual calculations when precision is critical.