Calculator Inputs
Example Data Table
| Series Example | Suggested Test | Key Input | Expected Result |
|---|---|---|---|
| ∑ 1 / n² | p-Series | p = 2 | Absolutely Convergent |
| ∑ (3/4)ⁿ | Geometric | r = 0.75 | Absolutely Convergent |
| ∑ 1 / n | p-Series | p = 1 | Divergent |
| ∑ n / 2ⁿ | Ratio | L < 1 | Absolutely Convergent |
| ∑ (-1)ⁿ / √n | Nth-Term + Absolute Check | |aₙ| = 1/√n | Not Absolutely Convergent |
Formula Used
Absolute convergence definition: A series ∑aₙ is absolutely convergent when ∑|aₙ| converges.
Ratio test: If L = lim |aₙ₊₁/aₙ|, then L < 1 implies absolute convergence, while L > 1 implies divergence.
Root test: If L = lim ⁿ√|aₙ|, then L < 1 implies absolute convergence, while L > 1 implies divergence.
p-series rule: The benchmark series ∑1/n^p converges when p > 1 and diverges when p ≤ 1.
Geometric rule: The series ∑arⁿ converges absolutely when |r| < 1.
Limit comparison: If lim |aₙ|/bₙ = c for a positive finite constant c, then ∑|aₙ| and ∑bₙ behave the same.
Nth-term check: If lim |aₙ| is not zero, the series cannot be absolutely convergent.
How to Use This Calculator
- Enter a series name so the output is easier to identify in exports.
- Choose the most suitable test, such as ratio, root, geometric, p-series, comparison, or nth-term.
- Add any known limit values, benchmark values, or p and r parameters.
- Paste sample absolute term values |aₙ| separated by commas to generate ratios, roots, and partial sums.
- Press the calculate button to show the result above the form.
- Review the result badge, explanation, computed table, and graph.
- Use the CSV and PDF buttons to export the visible results for classwork, reports, or review.
FAQs
1. What does absolute convergence mean?
A series is absolutely convergent when the series formed by taking absolute values of every term also converges. This is stronger than ordinary convergence and guarantees convergence of the original series.
2. Why does the calculator ask for absolute term values?
Absolute convergence depends on ∑|aₙ|, not directly on ∑aₙ. Using positive magnitudes lets the calculator estimate ratios, roots, and partial sums consistently.
3. When should I use the ratio test?
Use the ratio test for series involving factorials, exponentials, or products where consecutive terms simplify nicely. It is most helpful when the limit of |aₙ₊₁/aₙ| is easy to estimate.
4. When is the root test better?
The root test works well for terms raised to the nth power, such as expressions containing powers like cⁿ or nested exponent patterns. It focuses on lim ⁿ√|aₙ|.
5. Does a zero term limit prove convergence?
No. If |aₙ| tends to zero, the series may still diverge. The zero limit is necessary, but not sufficient, so another test is usually needed.
6. What does an inconclusive result mean?
It means the selected test could not prove convergence or divergence with the values provided. This often happens when a ratio or root limit is close to 1.
7. Can I use this for alternating series?
Yes, but enter the absolute values of the terms if you want to check absolute convergence. Alternating behavior alone only addresses conditional convergence, not absolute convergence.
8. Are the sample ratios and roots exact proofs?
No. They are numerical estimates based on the values you provide. Exact conclusions should rely on true limits or known benchmark behavior whenever possible.