Ratio Test Calculator – Series Convergence
Test series convergence using the ratio test (D'Alembert's criterion)
Table of Contents
How to Use
- Select the type of series you want to analyze
- Enter the required parameters for your series
- Click calculate to apply the ratio test
- View the convergence result and explanation
What is the Ratio Test?
The ratio test (also known as D'Alembert's criterion) is a method for determining whether an infinite series converges or diverges. It examines the limit of the ratio of consecutive terms.
For a series Σaₙ, calculate L = lim(n→∞) |aₙ₊₁/aₙ|. Then: if L < 1, the series converges absolutely; if L > 1, the series diverges; if L = 1, the test is inconclusive.
When to Use the Ratio Test
The ratio test is particularly effective for series involving:
- Factorials (n!)
- Exponentials (aⁿ)
- Products of factorials and exponentials
- Power series (to find radius of convergence)
Limitations
The ratio test is inconclusive (L = 1) for many important series:
- P-series (Σ1/n^p) - use p-series test instead
- Harmonic series (Σ1/n) - diverges
- Alternating harmonic series - use alternating series test
Common Examples
Geometric series Σrⁿ: L = |r|, converges if |r| < 1
Factorial series Σ1/n!: L = 0, converges
Exponential series Σxⁿ/n!: L = 0, converges for all x
Frequently Asked Questions
- What does it mean when the ratio test is inconclusive?
- When L = 1, the ratio test cannot determine convergence. You need to use another test such as the root test, comparison test, integral test, or alternating series test depending on the series structure.
- What's the difference between the ratio test and root test?
- Both tests examine similar limits but use different approaches. The ratio test looks at |aₙ₊₁/aₙ|, while the root test examines |aₙ|^(1/n). They often give the same result, but sometimes one is easier to compute than the other.
- Can the ratio test determine conditional convergence?
- No, the ratio test only determines absolute convergence. If L < 1, the series converges absolutely. For conditional convergence (converges but not absolutely), you need other tests like the alternating series test.
- Why does the ratio test work?
- The ratio test compares your series to a geometric series. If the ratio of consecutive terms approaches a value less than 1, the series behaves like a convergent geometric series. If greater than 1, it behaves like a divergent geometric series.