Absolutely convergent
The series of magnitudes Σ|aₙ| converges. This guarantees that the original series converges.
Series Convergence Calculator with Tests and Steps
Enter the nth term of an infinite series to test for absolute convergence, conditional convergence or divergence. The calculator applies supported symbolic tests, shows ratio and root diagnostics, and graphs partial sums—all locally in your browser.
Enter the nth term
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Convergence conclusion
Enter a series to begin.
The conclusion will appear here.
Several convergence tests will be checked.
Convergence tests and steps
Tests are applied conservatively. A sampled ratio, root or partial sum is clearly labeled as diagnostic when it is not a symbolic proof.
Test a series to see the nth-term condition and the most relevant convergence tests.
Partial-sum behavior
Partial sums can illustrate behavior but cannot by themselves prove convergence.
| Terms used | Last index | Last term | Partial sum |
|---|---|---|---|
| Partial-sum checkpoints will appear here. | |||
Calculator methodology and limits
- Developed and reviewed by
- Starlight Robotics calculator team
- Last reviewed
- Symbolic scope
- Recognizes supported algebraic power, rational, exponential/geometric, and alternating power behavior in one integer variable n.
- Test policy
- Symbolic results are labeled as such. Finite ratio, root and partial-sum samples are diagnostics; unsupported forms remain inconclusive rather than being guessed.
- Numeric limits
- Starting index 0–1,000,000, a 300-character expression, and 10–5,000 partial-sum terms using browser floating-point arithmetic.
- Important limit
- No finite list of partial sums proves convergence. An inconclusive result may require integral, comparison, Dirichlet, condensation or another analytic test.
- Reference
- Test statements follow OpenStax Calculus Volume 2, Chapter 5, including its ratio and root tests.
- Privacy
- Your formula, indices, results, graph and downloaded steps remain on your device.
How to test a series for convergence
- Write the nth term an, without the summation sign. For example, enter
1/n^2. - Set the first integer index. Use 1 for most p-series and 0 when the formula is defined there.
- Select Test convergence, then read the conclusion and whether it is symbolic, numerical evidence or inconclusive.
- Inspect the individual tests. Use the partial-sum graph only to understand behavior, not as a proof.
Core convergence rules
| Test | Conclusion |
|---|---|
| nth-term divergence | If lim aₙ ≠ 0 or does not exist, then Σaₙ diverges. A zero limit alone proves nothing. |
| p-series | Σ1/nᵖ converges for p > 1 and diverges for p ≤ 1. |
| Ratio test | For L = lim |aₙ₊₁/aₙ|, the series converges absolutely if L < 1 and diverges if L > 1; L = 1 is inconclusive. |
| Root test | For L = lim ⁿ√|aₙ|, the same thresholds apply as for the ratio test. |
| Alternating series | Σ(−1)ⁿbₙ converges when positive bₙ eventually decreases to 0. |
Conditionally convergent
The signed series converges through cancellation, while Σ|aₙ| diverges.
Divergent
A decisive test fails—for example, the terms do not approach zero—or a supported comparison is divergent.
Inconclusive
The available tests do not settle the expression. This is a valid mathematical outcome, not a convergence claim.
Worked examples
p-series
Σ1/n² has p=2>1, so it converges absolutely.
Harmonic series
Σ1/n has terms approaching zero but p=1, so it diverges.
Alternating harmonic
Σ(−1)ⁿ⁺¹/n converges by the alternating series test, but not absolutely.
Exponential decay
For Σn³/2ⁿ, polynomial growth loses to exponential decay; the ratio limit is 1/2, so the series converges absolutely.
Series convergence FAQs
Which convergence tests does this calculator use?
It starts with the nth-term condition, then recognizes supported geometric/exponential and p-series behavior, including limit-comparison and alternating forms. It also reports sampled ratio, root and partial-sum behavior.
Why does a zero term limit not prove convergence?
Terms must approach zero for a series to converge, but that condition is not enough. The harmonic series Σ1/n diverges even though 1/n → 0.
What does “symbolic conclusion” mean?
The calculator recognized a supported asymptotic form and applied a theorem with exact threshold logic. Numeric values shown beside ratio, root and partial-sum checks are still rounded browser calculations.
Why can the calculator return inconclusive?
No single test decides every series. Oscillatory, logarithmic, specially cancelling or unsupported terms may need a proof using a test outside this calculator’s symbolic scope.
Can partial sums prove that a series converges?
No. A long finite prefix can appear stable even when the infinite series diverges slowly. Partial sums are useful diagnostics only.
What syntax can I enter?
Use the variable n, ordinary arithmetic, parentheses and ^ for powers. Supported functions include exp, ln, log, sqrt, abs, and common trigonometric functions. Factorial notation is not currently supported.
Does my expression leave the browser?
No. The formula is parsed, tested, summed, graphed, copied and exported locally on your device.