The principle of contractive mapping is the most basic existence theorem in functional analysis. Through the analysis of typical examples of the limit of series for postgraduate entrance examination, this paper summarizes the general form of finding the limit series suitable for the principle of compression mapping, and shows the superiority of the principle of compression mapping in solving the limit of recursive mathematical series.

The principle of contractive mapping was put forward by the famous Polish mathematician Stefan Banach in 1922. It is the most commonly used ontology in the whole analytical science, and it is widely used.

For example, the existence theorem of implicit function and the existence and uniqueness of solutions of differential equations. This paper mainly studies the application of contractive mapping principle in the limit of sequence. Many references have talked about this application. On the basis of previous teaching experience, this paper systematically summarizes the application of contractive mapping principle in the limit of a kind of recursive sequence to further demonstrate its superiority.

1. Basic concepts and theorems

For the completeness of structure and the convenience of narration, we give several concepts and theorems in the literature.

Definition 1. 1< (x, ρ) is a metric space, and t is the mapping from x to x, if there is 0.

Theorem 1.2 (principle of contractive mapping) Let (x, ρ) be a complete distance space and T be a contractive mapping from X to X, then T has a unique fixed point on X, that is, there is a unique X? X, so tx = X.