Analyze function sequences with interval supremum estimates. Enter bounds, tolerances, and sample expressions for comparison. Get organized outputs for careful convergence checking and review.
Submit the form to see whether the entered supremum bounds support uniform convergence on the chosen interval.
Uniform convergence is stronger than pointwise convergence. It compares the whole interval at once. This calculator works with supremum errors or valid upper bounds. You enter indices and matching values for Mn. Each Mn represents the largest gap between fn(x) and the limit function f(x) over the chosen interval. The calculator then checks whether the tail of the sequence stays below a selected tolerance. That makes the result useful for fast screening, teaching, homework support, and numerical interpretation.
The key idea is simple. If sup |fn(x) - f(x)| goes to zero, then fn converges uniformly to f on that interval. In many problems, you do not compute the exact supremum. Instead, you build an upper bound Mn. If Mn also tends to zero, the conclusion still follows. This page supports both styles. That makes it practical for sequences built from powers, trigonometric terms, exponentials, rational functions, and bounded products.
The most important line is the suggested index N. When it appears, every entered value from that point onward is already at or below ε. That means the submitted tail supports uniform convergence at the chosen tolerance. The trend check also matters. A decreasing tail often signals improvement, but it is not a proof by itself. The result should be read as a numerical or bound-based test. Formal proofs still require correct mathematical justification.
This calculator is helpful when you already know the candidate limit and need a quick comparison of error bounds across several n-values. It is also useful for class examples. For instance, xn on [0, 1/2] has supremum error 2-n, which clearly tends to zero. On [0, 1], the same sequence does not converge uniformly to zero. A good bound table reveals that difference quickly. Use the article, formula section, example data, and FAQs below to interpret results correctly.
Uniform convergence test:
fn → f uniformly on I if
supx ∈ I |fn(x) - f(x)| → 0 as n → ∞
Bound-based version:
If supx ∈ I |fn(x) - f(x)| ≤ Mn and Mn → 0, then fn → f uniformly on I.
Tolerance check used by this page:
Find N such that Mn ≤ ε for every entered n ≥ N.
1. Enter labels for the function sequence and its limit.
2. Enter the interval endpoints a and b.
3. Choose whether your values are exact suprema or upper bounds.
4. Enter the n-values as a comma-separated list.
5. Enter the matching error bounds Mn in the same order.
6. Set a tolerance ε that you want the tail to satisfy.
7. Press Calculate to see the conclusion and tail summary.
8. Use CSV or PDF export to save the table.
Example: fn(x) = xn on [0, 1/2], with limit f(x) = 0. Here Mn = (1/2)n.
| n | Mn = sup |xn - 0| |
|---|---|
| 1 | 0.5 |
| 2 | 0.25 |
| 3 | 0.125 |
| 4 | 0.0625 |
| 5 | 0.03125 |
| 6 | 0.015625 |
| 7 | 0.0078125 |
| 8 | 0.00390625 |
Uniform convergence means the maximum difference between fn and f over the whole interval becomes arbitrarily small for large n.
Enter either the exact supremum error sup |fn - f| or any valid upper bound that is guaranteed to dominate that supremum on the interval.
No. It gives a numerical or bound-based check. A formal proof still needs correct reasoning about the sequence and interval.
Pointwise convergence tests each x separately. Uniform convergence controls every x on the interval at the same time.
The supremum depends on the interval. A sequence may converge uniformly on one interval but fail on a larger one.
It is the first entered index whose remaining tail values all stay at or below your chosen tolerance ε.
Yes. If your upper bounds Mn go to zero, they are enough to justify uniform convergence.
The result is inconclusive when the entered tail does not stay below ε or the values do not show a clear decreasing pattern.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.