Pointwise Convergence Calculator

Calculator Inputs

Sequence model

Choose a common sequence family with a known pointwise limit rule.

Domain start

Domain end

Sample point count

Ignored when custom sample points are provided below.

Custom sample points

Use commas, spaces, or semicolons between values.

Focus point for n-graph

Starting n

Ending n

n used on domain graph

Tolerance ε

Formula Used

Pointwise convergence means a sequence of functions f_n(x) approaches a limit function f(x) at each fixed x.

For every fixed x and every ε > 0, there exists N = N(x, ε) such that n ≥ N implies |f_n(x) − f(x)| < ε.

This calculator applies that idea numerically across sampled points and tested indices. It uses exact known limit rules for the selected family, then checks how quickly the sampled values approach that limit.

1. f_n(x) = xⁿ → 0 for |x| < 1, 1 at x = 1, and no finite limit otherwise.

2. f_n(x) = x / (n + 1) → 0 for every real x.

3. f_n(x) = sin(nx) / n → 0 for every real x.

4. f_n(x) = nx / (1 + nx²) → 0 at x = 0 and 1/x for x ≠ 0.

5. f_n(x) = x / (1 + nx²) → 0 for every real x.

6. f_n(x) = (1 + x/n)ⁿ → e^x for every real x.

The reported “First Stable N” is the first tested index whose remaining tested tail stays within ε for that sampled point.

How to Use This Calculator

Choose a sequence model from the list.
Enter a domain interval or provide custom sample points.
Set the starting and ending values of n.
Enter a tolerance ε for the convergence check.
Pick a focus point x to track convergence across n.
Press Analyze Convergence.
Read the summary cards, sampled point table, and graphs.
Use the CSV and PDF buttons to export the visible analysis.

Example Data Table

Example using f_n(x) = x / (n + 1), where the pointwise limit is 0 for all x.

x	f₅(x)	f₅₀(x)	Error at n = 50
-2	-0.333333	-0.039216	0.039216
-1	-0.166667	-0.019608	0.019608
0	0	0	0
1	0.166667	0.019608	0.019608
2	0.333333	0.039216	0.039216

Frequently Asked Questions

1. What does pointwise convergence mean?

Pointwise convergence means each fixed input x has its own convergence story. For every chosen x, the sequence values f_n(x) move toward a limit f(x) as n grows, but the required index N may vary from one point to another.

2. Does this calculator give a formal proof?

No. It gives numerical evidence for standard sequence families with known limit rules. The results help inspection, teaching, and intuition, but a formal proof still depends on mathematical argument rather than sampled computation alone.

3. Why can some sampled points converge while others do not?

Some function sequences behave differently at different locations. A classic example is xⁿ: it converges to 0 for |x| < 1, equals 1 at x = 1, and has no finite limit for many other points.

4. Why is the stable N different for different x values?

Pointwise convergence allows N to depend on x. Some points approach the limit quickly, while others need larger indices before the tail remains within the chosen tolerance ε.

5. What does the tolerance ε control?

The tolerance ε sets how close f_n(x) must be to the candidate limit before the calculator treats the tested tail as numerically stable.

6. How is pointwise convergence different from uniform convergence?

Pointwise convergence lets the required N vary with x. Uniform convergence is stronger because one single N must work for every point in the whole domain at the same time.

7. Why should I use custom sample points?

Custom points help you inspect important locations directly. They are useful near endpoints, singular points, oscillatory regions, or any values where you expect unusual convergence behavior.

8. Which model is best for learning the concept?

The model x / (n + 1) is easy for beginners because every point goes to 0. The model xⁿ is excellent for seeing how convergence can depend strongly on the chosen point.