Ratio Test Calculator
Ratio Test Calculator
Apply the Ratio Test to determine convergence of infinite series. Calculate L = lim(n→∞)|aₙ₊₁/aₙ| with step-by-step solutions.
Ratio Test Result
Ratio Test Rule:
If L < 1: Series converges absolutely. If L > 1: Series diverges. If L = 1: Test is inconclusive.
The ratio test compares successive terms to determine convergence behavior.
Step-by-Step Calculation:
Analysis Details:
The Ratio Test examines the limit of the ratio of successive terms to determine convergence.
What is the Ratio Test?
The Ratio Test is a convergence test for infinite series. Given a series ∑aₙ, we compute the limit L = lim(n→∞)|aₙ₊₁/aₙ|. The test states:
- If L < 1: The series converges absolutely
- If L > 1: The series diverges
- If L = 1: The test is inconclusive (series may converge or diverge)
The Ratio Test is particularly useful for series involving factorials, exponentials, and geometric-like terms.
Ratio Test Rules and Applications
Geometric Series
L = |r|
Converges if |r| < 1
Factorial Series
L = 0
Always converges
p-Series
L = 1
Test inconclusive
Exponential
L = 0
Converges ∀x
Ratio Test Formula
Decision Criteria:
• If L < 1: ∑aₙ converges absolutely
• If L > 1: ∑aₙ diverges
• If L = 1: Test fails (try another test)
Common Ratio Test Examples
| Series | General Term aₙ | Limit L | Conclusion |
|---|---|---|---|
| ∑(1/2)ⁿ | (1/2)ⁿ | 1/2 | Converges (geometric) |
| ∑1/n! | 1/n! | 0 | Converges absolutely |
| ∑n/2ⁿ | n/2ⁿ | 1/2 | Converges |
| ∑n! | n! | ∞ | Diverges |
| ∑1/n | 1/n | 1 | Test fails (diverges) |
| ∑1/n² | 1/n² | 1 | Test fails (converges) |
Step-by-Step Examples
Example 1: ∑(1/2)ⁿ
- General term: aₙ = (1/2)ⁿ
- Next term: aₙ₊₁ = (1/2)ⁿ⁺¹
- Compute ratio: |aₙ₊₁/aₙ| = |(1/2)ⁿ⁺¹/(1/2)ⁿ| = |1/2| = 1/2
- Take limit: L = lim(n→∞) 1/2 = 1/2
- Since L = 1/2 < 1, the series converges absolutely
- Conclusion: Geometric series with |r| = 1/2 < 1
Example 2: ∑1/n!
- General term: aₙ = 1/n!
- Next term: aₙ₊₁ = 1/(n+1)!
- Compute ratio: |aₙ₊₁/aₙ| = |(1/(n+1)!)/(1/n!)| = |n!/(n+1)!| = 1/(n+1)
- Take limit: L = lim(n→∞) 1/(n+1) = 0
- Since L = 0 < 1, the series converges absolutely
- Conclusion: Factorial denominator ensures convergence
When to Use Ratio Test
Best Applications:
- Factorial terms: Series involving n! in denominator
- Exponential terms: Series with rⁿ or xⁿ
- Geometric-like series: Terms with constant ratio pattern
- Series with powers: Terms like nᵏ * rⁿ
- Alternating series: With absolute convergence check
Limitations:
- p-series: Fails for ∑1/nᵖ (L = 1 for all p)
- Slow convergence: May be inconclusive for borderline cases
- Complex terms: May require simplification before applying
- Conditional convergence: Only tests absolute convergence
Related Convergence Tests
Frequently Asked Questions (FAQs)
Q: What does L = 1 mean in the Ratio Test?
A: When L = 1, the Ratio Test is inconclusive. The series may converge or diverge, and you need to use another test (like Comparison Test, Integral Test, or p-series test).
Q: Can the Ratio Test prove conditional convergence?
A: No, the Ratio Test only determines absolute convergence. For conditional convergence, you need tests like the Alternating Series Test.
Q: Why is the Ratio Test good for factorial series?
A: Because (n+1)! = (n+1)n!, so the ratio simplifies nicely: |aₙ₊₁/aₙ| = |something/(n+1)|, which often goes to 0.
Q: How accurate is this calculator?
A: The calculator uses symbolic simplification and limit calculation algorithms to provide accurate results for common series patterns. For complex series, it provides guidance on manual calculation.
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