1、Chapter 1 Functions and Limits,1.4 The Limits of Sequences,I. Infinite Sequence,is an ordered arrangement of real numbers.,Formal Definition Infinite sequence is a function whose domain is the set of positive integers and whose range is a set of real numbers.,For instance:,Q: Do they converge to 1?,
2、I. Infinite Sequence,Relationship with the two limits,?,II. Limit of Infinite Sequence,II. Limit of Infinite Sequence,Def: The sequence is said to converge to L, and we write if for each given number , there is a corresponding positive number N such that .A sequence that fails to converge to any fin
3、ite number L is said to diverge, or to be divergent.,II. Limit of Infinite Sequence,Geometric interpretation,When , all the points xn are in the neighborhood where there are finite points (at most N points) not in .,Example 1,Proof,Let be given.,that is,If we want ,We choose,That is,II. Limit of Inf
4、inite Sequence,Example 1,II. Limit of Infinite Sequence,1. Uniqueness,Th: Let,2. Boundedness,Th: Let,III. Properties of the Limit,1. Uniqueness,Th:,2. Local boundedness,Th:,III. Properties of the Limit,3. Conservation of the sign,Corollary:,Th:,Th:,III. Properties of the Limit,Th: If,3. The convergence of subsequence,III. Properties of the Limit,Review of the limit, , s.t,Exercises of the limit,Prove the following limits.,
