Calculator
Example Data Table
| Series model | Inputs | Expected result | Reason |
|---|---|---|---|
| 5(0.4)n-1 | a = 5, r = 0.4 | Convergent | |r| is less than 1. |
| 3/n2 | C = 3, p = 2 | Convergent | p is greater than 1. |
| 1/n | C = 1, p = 1 | Divergent | Harmonic series diverges. |
| (-1)n+1/n | C = 1, p = 1 | Conditionally convergent | Alternating terms shrink to zero. |
Formula Used
Geometric series: Σ arn-n0 converges when |r| < 1. The sum is a/(1-r).
p-series: Σ C/np converges when p > 1. It diverges when p ≤ 1.
Alternating p-series: Σ (-1)n-n0C/np converges when p > 0. It is absolute when p > 1.
Ratio test: L = lim |an+1/an|. It converges if L < 1. It diverges if L > 1.
Root test: L = lim ⁿ√|an|. It uses the same decision rule as the ratio test.
nth-term test: If lim an is not zero, the series diverges.
Limit comparison: For positive terms, a finite positive limit makes both series share the same behavior.
How to Use This Calculator
- Select the series model or convergence test.
- Enter the values needed for that test.
- Choose a starting index and term count for partial sums.
- Set decimal places and tolerance for limit decisions.
- Press Submit to show the result above the form.
- Use CSV or PDF buttons to save the output.
About Series Convergence Testing
Understanding Series Tests
A series adds infinitely many terms. Its total may settle near one value. It may also grow without a bound. This calculator gives a structured check. It does not replace algebra. It helps you organize common evidence.
Why Convergence Matters
Convergence is useful in probability, estimation, finance, physics, and numerical work. Many statistical models use infinite sums. A bad series choice can distort expected values. It can also break an approximation. A clear test protects later decisions.
Main Tests Included
The geometric test checks powers of a fixed ratio. The p-series test checks terms shaped like one over n raised to p. The alternating p-series test checks sign changes and shrinking terms. The ratio and root tests use limits from advanced calculus. The nth-term test is a fast warning. If terms do not approach zero, the series diverges.
Using Partial Sums
Partial sums add only a selected number of terms. They show early behavior. They do not prove convergence alone. Slow series can look stable too early. The calculator reports a partial sum and, when possible, an error estimate. Geometric, p-series, and alternating p-series models have useful remainder bounds.
Reading the Verdict
A convergence result means the infinite total exists. Absolute convergence means the positive version also converges. Conditional convergence means signs help the series settle. Divergence means no finite sum exists. Inconclusive means another test is needed. This is normal. No single test solves every series.
Best Practice
Start with the nth-term test. Then check for a known pattern. Use ratio or root tests when factorials, powers, or products appear. Use p-series ideas for rational powers of n. Use alternating tests when signs switch. Compare the answer with partial sums. Save the CSV for audit notes. Download the PDF for a quick report.
Limits of Numerical Work
A calculator can support judgment. It cannot see every symbolic pattern. Enter limits only after simplifying your expression. Use exact math when possible. Treat borderline cases carefully. Values near one in ratio or root tests need deeper analysis.
Common Input Tips
Keep the starting index consistent. Use positive term counts. Choose enough terms for review. Increase precision for small values. Check units before sharing reports safely with others later.
FAQs
What does convergence mean?
Convergence means the infinite sum approaches a finite value. The terms alone are not enough. The total of all terms must settle toward one number.
What does divergence mean?
Divergence means the series has no finite total. It may grow forever, oscillate, or fail another required condition.
Why can a zero term limit be inconclusive?
Every convergent series has terms approaching zero. Some divergent series also have terms approaching zero. The harmonic series is a common example.
When should I use the ratio test?
Use the ratio test for factorials, exponentials, products, and powers. It works best when a clean limit appears after simplification.
When should I use the root test?
Use the root test when the nth power appears in the general term. It is useful for expressions like powers raised to n.
What is conditional convergence?
Conditional convergence means the alternating series converges, but its absolute version diverges. The signs are essential to the final result.
Can partial sums prove convergence?
Partial sums show numerical behavior only. They can support a conclusion, but a formal test is needed for proof.
Why is my ratio test result inconclusive?
The ratio test is inconclusive when the limit equals one. You should try p-series, comparison, alternating, or another method.