Uniform Convergence Test Calculator

Calculator inputs

This tool combines interval bounds, sampled supremum values, and majorant or tail estimates. It is designed for fast checking, study, and report export.

Analysis mode

Choose whether you are testing a sequence or a function series.

Function family

Use the family that best matches your proof or exercise setup.

Preferred criterion

Automatic mode selects the bound style that fits the family.

Interval start a

Interval end b

Exponent p

Used in terms involving n^p.

Ratio r

Used only by the r^n cos(nx) family.

Starting n

Ending n

Sampling grid points

Epsilon target

The chart scan reports the first plotted n meeting this threshold.

Interpretation note: sequence mode checks whether the sup norm of the chosen family tends to zero on the interval. Series mode adds a majorant or tail estimate when the family supports it.

Example data table

Example scenario: series mode with u_n(x) = xⁿ / n² on [0, 0.8]. The M-test works because max |x| = 0.8 < 1.

n	Bound (0.8^n / n^2)	Approximate remaining tail
1	0.8000000000	0.2747946000
2	0.1600000000	0.1147946000
3	0.0568888889	0.0579057111
4	0.0256000000	0.0323057111
5	0.0131072000	0.0191985111
6	0.0072817778	0.0119167333

Formula used

Sequence criterion: sup_x∈I |f_n(x) - g(x)| → 0.

In this calculator, the default candidate limit for supported sequence families is g(x) = 0.

Weierstrass M-test: if |u_n(x)| ≤ M_n on the whole interval and ΣM_n converges, then Σu_n(x) converges uniformly.

Uniform Cauchy idea: estimate the tail sup_x∈I |Σ_k=N+1^∞ u_k(x)| and make it smaller than epsilon.

Selected family bound

For each chosen family, the tool builds a bound of the form sup |term| ≤ B_n.

The final decision is based on whether B_n or the tail estimate tends to zero.

How to use this calculator

Choose sequence or series mode to match your problem.
Pick the closest supported family.
Enter the interval endpoints exactly as in the theorem or exercise.
Set p or r when your family uses them.
Choose a plotted n-range wide enough to show decay clearly.
Use epsilon to see when the displayed bound first becomes small enough.
Read the verdict, then inspect the chart and data table.
Export the report as CSV or PDF for homework notes, proof drafts, or revision sheets.

FAQs

1) What does this calculator actually test?

It estimates uniform convergence using interval-wide bounds, sampled sup values, and majorant or tail formulas for supported families. It helps certify many standard textbook cases quickly.

2) Why is the supremum bound important?

Uniform convergence depends on one bound working for every x in the interval at once. The sup norm captures the worst-case error, not just behavior at isolated points.

3) When is the M-test decisive?

It is decisive when you can dominate every term by a positive sequence M_n independent of x, and the series ΣM_n converges.

4) Why can sampled maxima differ from analytic bounds?

Sampling only checks finitely many x-values. Analytic bounds cover the whole interval, so they are usually safer and often slightly larger than sampled maxima.

5) Can this tool prove failure of uniform convergence?

In several families, yes. For example, x^n on intervals with max |x| ≥ 1 does not decay uniformly to zero, and e^(-n x^2) fails on intervals containing 0.

6) What does the epsilon target mean?

It is your tolerance threshold. The calculator scans plotted n-values and reports the first one where the displayed bound or tail estimate drops below epsilon.

7) Is the graph itself a proof?

No. The graph is a visual aid. The proof comes from the interval-wide bound, the M-test, or the Cauchy tail estimate stated in the report.

8) Which interval choices are most delicate?

Endpoints near 1, -1, or 0 often matter most. Those points can prevent decay, break domination, or change whether a geometric or exponential majorant works.