数学专业英语翻译2-7

1、New Words & Expressions:assume 假定假定 sequence 序列，数列序列，数列converge 收敛收敛 series 级数，序列级数，序列diverge 发散发散 subscript 下标下标imaginary part 虚部虚部 succession 连贯性连贯性 imply 蕴含，推出蕴含，推出 successor 后继后继 recursion formula 递推公式递推公式2.7 序列及其极限序列及其极限Sequences and Their LimitsKey points: In everyday usage of the English

2、language, the words “sequence” and “series” are synonyms, and they are used to suggest a succession of things or events arranged in some order.7-A The definition of sequences在日常英语中，单词在日常英语中，单词“sequence” 和和 “series” 是同义是同义词，用以表示按某种次序排列的一串东西或事件。词，用以表示按某种次序排列的一串东西或事件。In mathematics these words have spe

3、cial technical meanings.在数学中这些单词有特殊的专业含义。在数学中这些单词有特殊的专业含义。The word “sequence” is employed as in the common use of the term to convey the idea of a set of things arranged in order, but the word “series” is used in a somewhat different sense.像通常用法一样，术语像通常用法一样，术语 “sequence” 用以表达按次序用以表达按次序排列的一串东西的意思，但是排

4、列的一串东西的意思，但是 “series” 一词则用于一词则用于表示别的意思（级数）。表示别的意思（级数）。The concept of a sequence will be discussed in this section, and series will be defined in section 11.序列的概念在本节讨论，级数的概念将在第序列的概念在本节讨论，级数的概念将在第11节定义。节定义。 If for every positive integer n there is associated a real or complex number an, then the order

5、ed set a1 , a2, , an , is said to define an infinite sequence. 如果对每一个正整数如果对每一个正整数 n都有一个实数或复数都有一个实数或复数an与之对与之对应应, 则有序集则有序集a1 , a2, , an , 称为一个无穷序列称为一个无穷序列. The important thing here is that each member of the set has been labeled with an integer so that we may speak of the first term a1, the second te

6、rm a2, and, in general, the nth term an. 这里重要的是集合中的每一个元素都由一个整数标这里重要的是集合中的每一个元素都由一个整数标 记，因此我们可以说第一项记，因此我们可以说第一项 , 第二项第二项, 一般的，第一般的，第n项项Each term has a successor and hence there is no “last ” term.每一项都有下面的一项，因此没有最后一项。每一项都有下面的一项，因此没有最后一项。The most common examples of sequences can be constructed if we gi

7、ve some rule or formula for describing the nth term. 如果我们给出描述第如果我们给出描述第n项的规则或者公式，就可以构项的规则或者公式，就可以构造出序列的常见例子。造出序列的常见例子。Thus, for example, the formula an=1/n defines a sequence whose first five terms are 1, 1/2, 1/3, 1/4, 1/5.例如，公式例如，公式an=1/n定义了一个序列，它的前五项是定义了一个序列，它的前五项是1, 1/2, 1/3, 1/4, 1/5.Sometimes

8、two or more formulas may be employed as, for example, a2n-1=1, a2n=2n2, the first few terms in this case being 1,2,1,8,1,18,1,32,1.有时可以使用两个或者多个公式有时可以使用两个或者多个公式(来定义一个序列来定义一个序列)，例如，例如， a2n-1=1, a2n=2n2，此时，前几项是，此时，前几项是Another common way to define a sequence is by a set of instructions which explains ho

9、w to carry on after a given start. Thus we may have a1=a2=1, an+1=an+an-1 for n 2 .另一种常用的定义序列的方法是，通过一串指令说明另一种常用的定义序列的方法是，通过一串指令说明在给定首项后如何给出后面的各项。在给定首项后如何给出后面的各项。This particular rule is known as a recursion formula and it defines a famous sequence whose terms are called Fibonacci numbers. The first f

10、ew terms are 1,1,2,3,5,8,13,21,34.这个特殊的规则就是常见的递推公式，它定义了一个这个特殊的规则就是常见的递推公式，它定义了一个著名的序列，其中的项称为菲波那契数。前几项著名的序列，其中的项称为菲波那契数。前几项是是.In any sequence the essential thing is that there be some function f defined on the positive integers such that f(n) is the nth term of the sequence for each n=1,2,3,对任意序列，最本质

11、的事就是存在某一个定义在正整对任意序列，最本质的事就是存在某一个定义在正整数集上的函数，使得对数集上的函数，使得对n=1,2,3,f(n) 为序列的第为序列的第n项。项。In fact, this is probably the most convenient way to state a technical definition of sequence. 事实上，这可能是给出序列专业定义的一种最方便的事实上，这可能是给出序列专业定义的一种最方便的方法。方法。 DEFINITION. A function f whose domain is the set of all positive in

12、tegers 1,2,3, is called an infinite sequence. The function value f(n) is called the nth term of the sequence.定义域为所有正整数集合的函数定义域为所有正整数集合的函数f 被称为无穷序列。被称为无穷序列。 函数值函数值f(n) 称为序列的第称为序列的第n项。项。The range of the function (that is, the set of function values) is usually displayed by writing the terms in order.

13、函数的值域（即是函数值的集合）通常以按顺序书写函数的值域（即是函数值的集合）通常以按顺序书写各项的方式来表示。各项的方式来表示。Very often the dependence on n is denoted by using subscripts, and we write an, xn, or something similar instead of f(n). Unless otherwise specified, all sequences in this chapter are assumed to have real or complex terms. 序列各项对序列各项对 n

14、的依赖性常利用下标来表示，记为的依赖性常利用下标来表示，记为an 或者其他类似的记号，而不记作或者其他类似的记号，而不记作f(n) 。除非另有说明，。除非另有说明，本章研究的序列都是假定具有实的项或复的项。本章研究的序列都是假定具有实的项或复的项。The main question we are concerned with here is to decide whether or not the terms f(n) tend to a finite limit as n increases infinitely. 7-B The limit of a sequence这里我们关心的主要问题

15、是当这里我们关心的主要问题是当n无限增加时，项无限增加时，项 f(n)是否会趋于一个有限的极限。是否会趋于一个有限的极限。To treat this problem, we must extend the limit concept to sequences. This is done as follows.为了解决这个问题，必须把极限的概念推广到序列中。为了解决这个问题，必须把极限的概念推广到序列中。做法如下：做法如下： DIFINITION. A sequence f(n) is said to have a limit L if, for every positive number e,

16、 there is another positive number N ( which may depend on e ) such that | anL | e for all n N . In this case, we say the sequence f(n) converges to L. A sequence which does not converge is called divergent.不收敛的序列被称为发散序列。不收敛的序列被称为发散序列。In this definition the function values f(n) and the limit L may be

17、 real or complex numbers. If f and L are complex, we may decompose them into their real and imaginary parts, say f=u+iv and L=a+ib.在这个定义中，函数值在这个定义中，函数值f(n) 和极限和极限L都可能是实数或者都可能是实数或者复数。如果复数。如果f和和L是复数是复数, 可以将其分解为实部和虚部。可以将其分解为实部和虚部。In other words, a complex-valued sequence f converges if and only if both

18、 the real part u and the imaginary part v converge separately. (P61 第三段最后一句第三段最后一句)换句话说，复值序列换句话说，复值序列f收敛当且仅当实部和虚部收敛当且仅当实部和虚部(序列序列)分别收敛。分别收敛。It is clear that any function defined for all positive real x may be used to construct a sequence by restricting x to take only integer values.显然显然, 每一个对所有正实数每一个对所有正实数 x 有定义的函数都可以用有定义的函数都可以用来构造一个序列，其办法是限制来构造一个序列，其办法是限制 x 只取整数值。只取整数值。The phrase “convergent sequence” is used only for a sequence whose limit is finite. A sequence with an infinite limit is said to diverge. There are, of cours