According to the convergence theorem, also known as Dirichlet convergence theorem; The conclusion of the theorem is: at the continuous point x of f(x), the series converges to f (x); At the discontinuous point X of f(x), the series converges to (f(x+0)+f(x-0))/2, that is, the average of the left and right limits of f(x) at the discontinuous point;

Defined in a way similar to sequence convergence. Cauchy Convergence Criterion: On the Convergence Definition of Function f(x) at Point x0. For any real number b>0, c > exists; 0, for any x 1, x2 satisfies 0.

Convergence and divergence of iterative algorithm

1, global convergence

For any X0∈[a, b], the sequence of points generated by the iterative formula Xk+ 1=φ(Xk) converges, that is, when k→∞, the limit of Xk tends to X*, then Xk+ 1=φ(Xk) is called in [a, b].

2. Local convergence

If X* exists in a neighborhood, r = {x || x-x *|| < δ}}, for any X0∈R, the convergence listed in Xk+ 1=φ(Xk) is called Xk+ 1=φ(Xk) converges to x *.