數(shù)學(xué)專業(yè)英語翻譯2-7

1、New Words & Expressions:assume 假定假定 sequence 序列，數(shù)列序列，數(shù)列converge 收斂收斂 series 級(jí)數(shù)，序列級(jí)數(shù)，序列diverge 發(fā)散發(fā)散 subscript 下標(biāo)下標(biāo)imaginary part 虛部虛部 succession 連貫性連貫性 imply 蘊(yùn)含，推出蘊(yùn)含，推出 successor 后繼后繼 recursion formula 遞推公式遞推公式2.7 序列及其極限序列及其極限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在日常英語中，單詞在日常英語中，單詞“sequence” 和和 “series” 是同義是同義詞，用以表示按某種次序排列的一串東西或事件。詞，用以表示按某種次序排列的一串東西或事件。In mathematics these words have spe

3、cial technical meanings.在數(shù)學(xué)中這些單詞有特殊的專業(yè)含義。在數(shù)學(xué)中這些單詞有特殊的專業(yè)含義。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.像通常用法一樣，術(shù)語像通常用法一樣，術(shù)語 “sequence” 用以表達(dá)按次序用以表達(dá)按次序排列的一串東西的意思，但是排

4、列的一串東西的意思，但是 “series” 一詞則用于一詞則用于表示別的意思（級(jí)數(shù)）。表示別的意思（級(jí)數(shù)）。The concept of a sequence will be discussed in this section, and series will be defined in section 11.序列的概念在本節(jié)討論，級(jí)數(shù)的概念將在第序列的概念在本節(jié)討論，級(jí)數(shù)的概念將在第11節(jié)定義。節(jié)定義。 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. 如果對(duì)每一個(gè)正整數(shù)如果對(duì)每一個(gè)正整數(shù) n都有一個(gè)實(shí)數(shù)或復(fù)數(shù)都有一個(gè)實(shí)數(shù)或復(fù)數(shù)an與之對(duì)與之對(duì)應(yīng)應(yīng), 則有序集則有序集a1 , a2, , an , 稱為一個(gè)無窮序列稱為一個(gè)無窮序列. 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. 這里重要的是集合中的每一個(gè)元素都由一個(gè)整數(shù)標(biāo)這里重要的是集合中的每一個(gè)元素都由一個(gè)整數(shù)標(biāo) 記，因此我們可以說第一項(xiàng)記，因此我們可以說第一項(xiàng) , 第二項(xiàng)第二項(xiàng), 一般的，第一般的，第n項(xiàng)項(xiàng)Each term has a successor and hence there is no “l(fā)ast ” term.每一項(xiàng)都有下面的一項(xiàng)，因此沒有最后一項(xiàng)。每一項(xiàng)都有下面的一項(xiàng)，因此沒有最后一項(xiàng)。The most common examples of sequences can be constructed if we gi

7、ve some rule or formula for describing the nth term. 如果我們給出描述第如果我們給出描述第n項(xiàng)的規(guī)則或者公式，就可以構(gòu)項(xiàng)的規(guī)則或者公式，就可以構(gòu)造出序列的常見例子。造出序列的常見例子。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定義了一個(gè)序列，它的前五項(xiàng)是定義了一個(gè)序列，它的前五項(xiàng)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.有時(shí)可以使用兩個(gè)或者多個(gè)公式有時(shí)可以使用兩個(gè)或者多個(gè)公式(來定義一個(gè)序列來定義一個(gè)序列)，例如，例如， a2n-1=1, a2n=2n2，此時(shí)，前幾項(xiàng)是，此時(shí)，前幾項(xiàng)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 .另一種常用的定義序列的方法是，通過一串指令說明另一種常用的定義序列的方法是，通過一串指令說明在給定首項(xiàng)后如何給出后面的各項(xiàng)。在給定首項(xiàng)后如何給出后面的各項(xiàng)。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.這個(gè)特殊的規(guī)則就是常見的遞推公式，它定義了一個(gè)這個(gè)特殊的規(guī)則就是常見的遞推公式，它定義了一個(gè)著名的序列，其中的項(xiàng)稱為菲波那契數(shù)。前幾項(xiàng)著名的序列，其中的項(xiàng)稱為菲波那契數(shù)。前幾項(xiàng)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,對(duì)任意序列，最本質(zhì)

11、的事就是存在某一個(gè)定義在正整對(duì)任意序列，最本質(zhì)的事就是存在某一個(gè)定義在正整數(shù)集上的函數(shù)，使得對(duì)數(shù)集上的函數(shù)，使得對(duì)n=1,2,3,f(n) 為序列的第為序列的第n項(xiàng)。項(xiàng)。In fact, this is probably the most convenient way to state a technical definition of sequence. 事實(shí)上，這可能是給出序列專業(yè)定義的一種最方便的事實(shí)上，這可能是給出序列專業(yè)定義的一種最方便的方法。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.定義域?yàn)樗姓麛?shù)集合的函數(shù)定義域?yàn)樗姓麛?shù)集合的函數(shù)f 被稱為無窮序列。被稱為無窮序列。 函數(shù)值函數(shù)值f(n) 稱為序列的第稱為序列的第n項(xiàng)。項(xiàng)。The range of the function (that is, the set of function values) is usually displayed by writing the terms in order.

13、函數(shù)的值域（即是函數(shù)值的集合）通常以按順序書寫函數(shù)的值域（即是函數(shù)值的集合）通常以按順序書寫各項(xiàng)的方式來表示。各項(xiàng)的方式來表示。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. 序列各項(xiàng)對(duì)序列各項(xiàng)對(duì) n

14、的依賴性常利用下標(biāo)來表示，記為的依賴性常利用下標(biāo)來表示，記為an 或者其他類似的記號(hào)，而不記作或者其他類似的記號(hào)，而不記作f(n) 。除非另有說明，。除非另有說明，本章研究的序列都是假定具有實(shí)的項(xiàng)或復(fù)的項(xiàng)。本章研究的序列都是假定具有實(shí)的項(xiàng)或復(fù)的項(xiàng)。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這里我們關(guān)心的主要問題

15、是當(dāng)這里我們關(guān)心的主要問題是當(dāng)n無限增加時(shí)，項(xiàng)無限增加時(shí)，項(xiàng) f(n)是否會(huì)趨于一個(gè)有限的極限。是否會(huì)趨于一個(gè)有限的極限。To treat this problem, we must extend the limit concept to sequences. This is done as follows.為了解決這個(gè)問題，必須把極限的概念推廣到序列中。為了解決這個(gè)問題，必須把極限的概念推廣到序列中。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.不收斂的序列被稱為發(fā)散序列。不收斂的序列被稱為發(fā)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.在這個(gè)定義中，函數(shù)值在這個(gè)定義中，函數(shù)值f(n) 和極限和極限L都可能是實(shí)數(shù)或者都可能是實(shí)數(shù)或者復(fù)數(shù)。如果復(fù)數(shù)。如果f和和L是復(fù)數(shù)是復(fù)數(shù), 可以將其分解為實(shí)部和虛部?？梢詫⑵浞纸鉃閷?shí)部和虛部。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)換句話說，復(fù)值序列換句話說，復(fù)值序列f收斂當(dāng)且僅當(dāng)實(shí)部和虛部收斂當(dāng)且僅當(dāng)實(shí)部和虛部(序列序列)分別收斂。分別收斂。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.顯然顯然, 每一個(gè)對(duì)所有正實(shí)數(shù)每一個(gè)對(duì)所有正實(shí)數(shù) x 有定義的函數(shù)都可以用有定義的函數(shù)都可以用來構(gòu)造一個(gè)序列，其辦法是限制來構(gòu)造一個(gè)序列，其辦法是限制 x 只取整數(shù)值。只取整數(shù)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