介绍序列的定义和特征属性,每一句话都是重点

1.definition

A sequence is a list of number like a_1,a_2,cdots,a_n in a given order,and each element of it is called term,and the integer n is called the index of a_n ,moreover,an infinite sequence of numbers is a function whose domain is the set of positive integers.

Sequences can be described by writing rules that specify their terms ,such as a_n=sqrt{n},b_n=(-1)^{n+1}frac{1}{n}

or by listing terms like begin{Bmatrix}{a_n}end{Bmatrix}=begin{Bmatrix} sqrt{1},sqrt2,sqrt3,cdots,sqrt nend{Bmatrix} ,and sometimes wirte begin{Bmatrix}a_nend{Bmatrix}=begin{Bmatrix} sqrt nend{Bmatrix}_{n=1}^{infty} ,and also it can be expressed as imgs.

e.g.

2.收敛和发散对应定义

Sometimes the numbers in sequence may approach a single number like:begin{Bmatrix} 1,frac{1}{2},frac{1}{3},cdots,frac{1}{n}end{Bmatrix} approaches to 0 as n gets large.use delta-sigma language,we get:

收敛->定值

ps:This definition is pretty similar to the definition of a function f(x) as x tends to infty. So we can analogy them

这里最好复习一下sigma-delta语言和对应的极限证明方法 ，下面是发散：

发散->infty

3.计算法则

同于一般极限情况，遵循[下图]：1.四则运算规律 2.三明治定理 3.连续性定理 4.洛必达法则

4.几个常见极限 star

5.有界单调的定义

and we can know [no proof]: