Calculator inputs
This estimator samples the chosen domain, checks late-sequence errors against your tolerance, and approximates the size of the exceptional set.
Preset guide
| Preset | Formula | Use case |
|---|---|---|
| Power sequence: x^n | f_n(x) = x^n, target f(x) = 0 | Classic example that tends to 0 on most points in [0,1], except near x = 1. |
| Oscillatory decay: sin(β n x) / n^α | f_n(x) = sin(β n x) / n^α, target f(x) = 0 | Oscillation remains, but amplitude shrinks with n^α. |
| Shrinking interval indicator | f_n(x) = 1{|x-c| ≤ 1 / n^α}, target f(x) = 0 | Useful for studying exceptional sets concentrated near one center point. |
| Reciprocal spike: 1 / (1 + n|x-c|^α) | f_n(x) = 1 / (1 + n|x-c|^α), target f(x) = 0 | Produces a narrowing spike around the chosen center. |
| Shifted linear decay: (x-c) / n^α | f_n(x) = (x-c) / n^α, target f(x) = 0 | Simple sequence with uniform decay away from scale issues. |
| Exponential decay: e^{-n|x-c|^α} | f_n(x) = e^{-n|x-c|^α}, target f(x) = 0 | Sharp decay away from the center with a possible exceptional point at c. |
Example data table
Example using the power sequence fn(x) = xn on [0,1] with target 0. The point x = 1 stays exceptional in the sampled table.
| x | f5(x) | f20(x) | Target | Observation |
|---|---|---|---|---|
| 0.00 | 0.00000 | 0.00000 | 0 | Already equal to the target. |
| 0.25 | 0.00098 | 0.00000 | 0 | Rapid decay toward zero. |
| 0.50 | 0.03125 | 0.00000 | 0 | Tail errors shrink quickly. |
| 0.75 | 0.23730 | 0.00317 | 0 | Converges more slowly near 1. |
| 1.00 | 1.00000 | 1.00000 | 0 | Sampled exceptional point remains nonconvergent. |
Formula used
1. Uniform sampling of the domain
xi = a + i(b-a)/(m-1), for i = 0, 1, ..., m-1.
2. Tail error at a sampled point
T(xi) = maxn = Ntail, ..., Nmax |fn(xi) - f(xi)|.
3. Sampled convergence test
xi is marked convergent when T(xi) ≤ ε.
4. Estimated exceptional share
s = (# nonconvergent sampled points) / m.
5. Estimated exceptional measure
μest ≈ s × |b-a|.
6. Decision rule
The tool reports estimated almost-everywhere convergence when the sampled exceptional share stays below your chosen threshold. This is a numerical heuristic, not a formal proof.
How to use this calculator
- Choose a preset sequence family that matches the convergence pattern you want to study.
- Enter the domain start and domain end values for the interval you want sampled.
- Select how many sample points should approximate the underlying measure on the interval.
- Set minimum n, maximum n, and the tail window to define the late sequence behavior.
- Choose ε as the tolerated late-stage error at each sampled point.
- Set the exceptional-set threshold percentage that you are willing to treat as negligible.
- Adjust α, β, and c when the preset uses exponents, oscillation, or a center point.
- Press the calculate button and review the verdict, summary cards, graph, and computed table.
- Use the CSV export for raw sampled data and the PDF export for a compact report.
Frequently asked questions
1. What does almost everywhere mean here?
It means failure may occur on a set of measure zero. This page approximates that idea by checking how many sampled points still violate the tail tolerance on your chosen interval.
2. Is this calculator a formal proof?
No. It is a numerical estimator. It helps explore sampled behavior, exceptional regions, and tolerance sensitivity, but analytic proofs still require rigorous mathematics beyond sampled evidence.
3. Why does the calculator use a tail window?
Almost-everywhere convergence concerns late behavior. The tail window focuses attention on the final sequence terms, reducing the influence of early values that may be far from the limit.
4. How should I choose ε?
Pick ε according to the precision you need. Smaller values make the test stricter, often enlarging the sampled exceptional set. Larger values are more forgiving but may hide slow convergence.
5. Why can a single sampled point change the verdict?
When the sample count is small, one point represents a larger share of the interval. Increasing the number of sampled points usually gives a finer approximation of the exceptional set.
6. What is the exceptional-set threshold percentage?
It is your practical cutoff for treating sampled nonconvergence as negligible. A smaller threshold makes the estimator stricter; a larger threshold tolerates broader sampled failure.
7. Which preset is best for a classic almost-everywhere example?
The reciprocal spike, shrinking indicator, and exponential decay presets are useful. They often create one concentrated troublesome location while most other sampled points move toward zero.
8. What do the CSV and PDF downloads contain?
The CSV includes the full sampled table with errors and convergence flags. The PDF summarizes the verdict, main metrics, and a compact preview of sampled rows.