Math Applets

Convergence of a sequence

Convergence:

Let (an)n∈N(a_n)_{n \in \mathbb{N}} be a sequence of real numbers. The sequence (an)n∈N(a_n)_{n \in \mathbb{N}} is said to be convergent towards a∈Ra \in \mathbb{R}, if for every ε>0\varepsilon > 0 there exists an N∈NN \in \mathbb{N} such that for all n∈Nn \in \mathbb{N} with n≥Nn \geq N it holds that ∣an−a∣<ε.|a_n - a| < \varepsilon. In this case, aa is called the limit of the sequence (an)n∈N(a_n)_{n \in \mathbb{N}}. We write lim⁡n→∞an=aoran→a for n→∞.\lim_{n \to \infty} a_n = a \quad \text{or} \quad a_n \to a \text{ for } n \to \infty.

How to use the applet

  • ∣an−a∣<εfor all n≥N|a_n-a|<\varepsilon \quad \text{for all } n\ge N for the sequence an=1na_n=\frac1n and the limit a=0a=0.

  • Drag the green point ε\varepsilon up/down to choose an error tolerance.

  • The two green dashed lines show the ε\varepsilon-neighborhood of the limit: a−εanda+ε(a=0).a-\varepsilon \quad \text{and} \quad a+\varepsilon \qquad (a=0).

  • The purple vertical line marks the computed threshold Nε=⌈1ε⌉N_\varepsilon=\left\lceil\frac1\varepsilon\right\rceil.

  • Red points correspond to indices n<Nεn < N_\varepsilon: the inequality ∣an−a∣<ε|a_n-a|<\varepsilon may fail.

  • Purple points correspond to indices n≥Nεn \ge N_\varepsilon: all of them lie inside the green band, i.e. ∣1n−0∣<ε.\left|\frac1n-0\right|<\varepsilon.


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  • Points shaped like <> act as sliders

Description

Link to code.

Reference: Lecture Notes Calculus 1 ( May22,2022 ) Definition 2.2 page 15

For the example i’ve selected the sequence (1n)n∈N(\frac{1}{n})_{n \in \mathbb{N}} with limit 0.

The epsilon neighborhood from the limit 00 is adjustable.

The index NεN_{\varepsilon} is calculated, when epsilon changes, by ∣1n−0∣<ε  ⟺   n≥1ε|\frac{1}{n} -0| < \varepsilon \iff \ n \ge \frac{1}{\varepsilon} giving Nε=⌈1ε⌉N_{\varepsilon}= \lceil \frac{1}{\varepsilon} \rceil .

    function findN(epsilon: number): number {
        return Math.ceil(1 / epsilon);
    }

The points are created from a loop:

    for (let n = 1; n <= maxPoints; n++) {
        const an = sequence(n);
        const isAfterN = n >= N;
        const point = board.create('point', [n, an], {
            name: '',
            size: 3,
            fillColor: isAfterN ? '#9C27B0' : '#F44336',
            strokeColor: isAfterN ? '#6A1B9A' : '#C62828',
            fixed: true,
        });
        sequencePoints.push(point);
    }