What is the meaning of the word convergence?

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What is the meaning of the word convergence?​

con·ver·gence | \\kən-ˈvər-jən(t)s \\. 1 : the act of converging and especially moving toward union or uniformity the convergence of the three rivers especially : coordinated movement of the two eyes so that the image of a single point is formed on corresponding retinal areas. 2 : the state or property of being convergent.
How do you know if the convergence is linear?
We say the convergence is linear if there exists r ∈ (0,1) r ∈ (0, 1) such that ∥xn+1 −x∞∥ ∥xn −x∞∥ ≤ r ‖ x n + 1 − x ∞ ‖ ‖ x n − x ∞ ‖ ≤ r for all n n sufficiently large. 2.2.1.1 Example The simple sequence xn =1 +(1 2)n x n = 1 + (1 2) n converges linearly to x∞ =1 x ∞ = 1 because

What are the three types of rates of convergence?​

There are three rates of convergence that we will focus on here—linear, superlinear, and quadratic—which are ordered from slowest to fastest. In our context, rates of convergence are typically determined by how much information about the target function f f we use in the updating process of the algorithm.
What is quadratic convergence?
Quadratic convergence is the fastest form of convergence that we will discuss here and is generally considered desirable if possible to achieve. We say the sequence converges at a quadratic rate if there exists some constant 0 < M < ∞ 0 < M < ∞ such that ∥xn+1 −x∞∥ ∥xn −x∞∥2 ≤ M ‖ x n + 1 − x ∞ ‖ ‖ x n − x ∞ ‖ 2 ≤ M for all n n sufficiently large.

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