Calculation result
Interpretation
Step-by-step solution
No calculation yet.
Formula used
The ratio test compares consecutive absolute terms. It uses a limiting ratio. Values below one prove absolute convergence.
The root test examines the nth root. A limsup handles oscillating sequences. Values above one prove divergence.
How to use the calculator
Enter the general term using n. Choose a testing mode. Add parameter values when needed.
Adjust iterations for difficult numerical limits. Review stability before trusting estimates. Exact proofs may need another test.
Power-series mode estimates radius and endpoints. Parameter mode scans a selected range. Export results after checking warnings.
Worked example data
| Series term | Suggested test | Typical limit | Expected conclusion |
|---|---|---|---|
| 1/n! | Ratio | 0 | Absolutely convergent |
| n²/3ⁿ | Ratio | 1/3 | Absolutely convergent |
| (n/(n+1))^(n²) | Root | e⁻¹ | Absolutely convergent |
| 1/n | Either | 1 | Inconclusive |
| (1/3)ⁿ | Either | 1/3 | Absolutely convergent |
Important limitations
A numerical limit is supporting evidence only. Slow convergence can mislead estimates. Borderline values require stronger analysis.
The ratio and root tests prove absolute convergence. They rarely settle conditional convergence. Endpoint decisions remain separate problems.
Undefined terms may require a later start. Complex terms use absolute magnitudes. Always verify delicate conclusions independently.
Frequently asked questions
What does a limit below one mean?
It proves the series converges absolutely. Both tests give this conclusion. No endpoint exception applies here.
What happens when the limit equals one?
The selected test is inconclusive. The series may still converge. Another convergence test is required.
Which test suits factorial expressions?
The ratio test is usually simpler. Consecutive factorials cancel cleanly. The calculator detects this pattern.
Which test suits powers raised to n?
The root test often simplifies them. Nth roots remove exponent structures. Automatic mode checks this pattern.
Can it test complex series?
Yes, magnitudes are tested numerically. Absolute convergence uses complex modulus. Symbolic simplification may remain limited.
Can it find a convergence radius?
Power-series mode estimates the radius. It also checks both endpoints. Numerical endpoint checks need verification.
Why does the calculator show instability?
Tail values may still be changing. More iterations can improve estimates. Oscillation may require limsup reasoning.
Does divergence require terms growing?
No, terms can shrink too slowly. A nonzero term limit proves divergence. Ratio values above one also prove it.
Which alternative test should I use?
Use comparison for similar positive terms. Use alternating tests for sign changes. Integral tests suit smooth positive terms.