Calculator inputs
Example data table
| Method | aₙ | bₙ | Known behavior of Σbₙ | n range | Observed relation | Expected outcome |
|---|---|---|---|---|---|---|
| Direct | 1 / (n² + n) | 1 / n² | Convergent | 2 to 30 | aₙ ≤ bₙ | Supports convergence |
| Limit | (3*n + 1) / (n³ + 4) | 1 / n² | Convergent | 5 to 40 | aₙ / bₙ approaches 3 | Supports convergence |
| Direct | 1 / n | 1 / (2*n) | Divergent | 2 to 40 | aₙ ≥ bₙ | Supports divergence |
Formula used
Direct comparison test: for positive terms, if 0 ≤ aₙ ≤ bₙ eventually and Σbₙ converges, then Σaₙ converges. If 0 ≤ bₙ ≤ aₙ eventually and Σbₙ diverges, then Σaₙ diverges.
Limit comparison test: if
L = lim (aₙ / bₙ), with 0 < L < ∞, then Σaₙ and Σbₙ have the same convergence behavior.
Numerical implementation: this calculator samples terms over a chosen interval, computes partial sums, checks tail inequalities, and estimates ratio stabilization from the selected tail segment.
How to use this calculator
- Enter the target sequence term
aₙusing valid mathematical syntax. - Enter a comparison sequence term
bₙthat you already understand well. - Select whether
Σbₙis known to converge or diverge. - Choose direct comparison when you expect an eventual inequality, or limit comparison when you expect matching asymptotic size.
- Set the start value, total sample size, and tail length.
- Submit the form and read the verdict, graph, partial sums, and detailed numeric table.
- Export the computed table to CSV or save the result section as PDF.
FAQs
1. What does this comparison test calculator evaluate?
It evaluates positive-term series numerically using direct comparison or limit comparison. It samples terms, checks tail inequalities, estimates tail ratios, and summarizes evidence for convergence or divergence.
2. Does the calculator provide a formal proof?
No. It provides numerical evidence from sampled terms. Your final proof still depends on choosing a valid comparison series and confirming the theorem’s conditions hold eventually.
3. What expression syntax is supported?
Use numbers, n, parentheses, + - * / ^, and functions such as ln(), log(), sqrt(), exp(), and abs(). Write multiplication explicitly, such as 3*n.
4. When should I use direct comparison?
Use direct comparison when you can reasonably expect one sequence to stay above or below the other after some large index. It is especially useful with simple power, logarithmic, and geometric bounds.
5. When is limit comparison better?
Use limit comparison when the two sequences have similar growth or decay, so the ratio aₙ/bₙ tends toward a positive finite constant. This is common with rational and mixed-power terms.
6. Why do I need the known behavior of Σbₙ?
The theorem needs a comparison series whose convergence or divergence is already known. The calculator uses your selection to translate a successful inequality or ratio test into a conclusion for the target series.
7. Why might the result be inconclusive?
The sampled tail may not show a stable inequality, the ratio may not settle, or one series may contain non-positive terms. Changing the comparison series or moving to larger n often helps.
8. Can I export the result?
Yes. The page includes CSV export for the computed numeric table and PDF export for the visible result section, including the summary, metrics, and plotted graph.