Convergence Order of a Numerical Method

Topics: Numerical Analysis - Numerical Method - Limit of a Sequence

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In many numerical methods, we calculate the first $n$ terms of a sequence to observe a possible convergence to the final answer.

A numerical method converges if:

$$|x_n-x_{n-1}|<|x_{n-1}-x_{n-2}|$$

That is, a numerical method converges if the difference between the last two approximations is always greater than or equal than the absolute error. Notice that this difference is a bound for the absolute error.