微积分和数学分析引论节选外文翻译资料

2023-01-07 15:26:50

本科毕业设计（论文）

外文翻译

微积分和数学分析引论节选

作者：R.柯朗，F.约翰

国籍：德国

出处：Introduction to Calculus and Analysis

中文译文：

1.7再论极限的概念

a.收敛和发散的定义

从1.6节讨论的那些实例，我们抽象出一下一般的极限概念：

假设对于给定的无穷点列，，，，存在一个数l，使得每一个包含点l的开区间（无论多么小），都包含着除去最多为有限个点以外的所有点。这时，数l称为序列，，的极限，或者我们说，序列，，是收敛的并且收敛于l，记作。

下述的极限定义是与此等价的：

对于任何正数ε（无论多么小），我们都能找到足够大的整数N=N（ε），使得从下标N以后[也就是说，对于ngt; N（ε）]，总有。

当然，作为一般的规律，容许界限ε之值越小，N（ε）必须选得越来越大；换句话说，当ε趋向于零时，N（ε）通常将要无限地增大。关于极限的这种笼统而直观的概念，使我们想象到将变得越来越靠近于l。这一图像在这里可由下述严格的“定性的”定义所代替：l的任何领域都包含着除去最多为有限个点以外的所有点。

显然，序列，，的极限l不能多于一个。因为，如果有两个不同的数l和lrsquo;，是同一序列，，的极限，我们就能划出分别包含点l和lrsquo;而又不相重叠的两个开区间。因为每一个区间都包含除有限个点之外所有的点，所以序列不能是无限的。因此，收敛的序列的极限是唯一确定的。

另一个明显而又有用的推论是：如果我们从收敛的序列中去掉任何一些项，则所得序列与原序列收敛于同样的极限。

一个序列，如果不收敛，则称为发散的。如果当n增加时，数增大而超过任何正数，我们就说序列发散于；正如前面已经提到过的，这时我们记为。类似地，如果当n增加时，数在正值方向上增大而超过任何界限，我们就记为。但是，发散性也可按另一种方式出现，例如对于序列=-1，= 1，=-1，= 1，，其各项在两个不同的数值上来回摆动。

显然，去掉有限多项，既不会影响序列的发散性，也不会影响其收敛性。

对序列，，，如果存在一个有限区间包含其所有点，则称此序列为有界的。任何有限区间都包含在某一个以原点为中心的有限区间之中。因此，序列是有界的这一要求，指的就是存在着一个数M，使得对于一切n，都有。

收敛的序列，，必定是有界的。因为设l是此序列的极限并取ε=1，我们由收敛性的定义可知，从某一个N以后所有的都位于以l为中心、长度为2的区间中。在此序列中可能位于此区间之外的那些项，只有。然而，这时我们可以找到一个更大的有限区间，使得它还包含。

b.极限的有理运算

由极限的定义立即得知，我们可以按照下述规则进行极限的加法、乘法、减法和除法等初等运算。

如果序列，，的极限是a，序列的极限是b，则序列=的极限是c，并且

序列=也是收敛的，并且

类似地，序列也是收敛的，并且

如果极限b不为零，则序列也是收敛的，并且具有极限

总而言之，我们可以将有理运算同求极限的过程交换次序；无论是先求极限然后进行有理运算，还是先进行有理运算然后求极限，我们将得到同样的结果。

只要证明了这些法则之一，所有这些法则的证明也就很容易做到了。我们来看极限的乘法。如果关系式成立，则对于任何正数ε，只要我们将n选得充分大，比如n〉N（ε），便可保证

和

如果我们写出

并且考虑到存在与n无关的正数M，使得，则得到

因为如果我们把ε选得足够小，就可使成为任意小量，所以对于一切充分大的n值，和之差实际上将会成为任意小；这正是下列等式所要求的论述：

仿照这个例子，读者可以证明其余有理运算的法则。

借助于这些法则，许多极限都能很容易地算出。例如，我们有

，

因为在第二个表达式中，我们可以直接对分子和分母取极限。

下述简单法则也是经常会用到的：如果，，并且对于每一个n都有，则。然而，我们决不能指望a总是大于b，正如序列，所表明的，对于这两个序列，有a=b=0。

附：外文原文

1.7 Discussion of the Concept of Limit

a. Definition of Convergence and divergence

From the examples discussed in Section 1.6 we abstract the following general concept of limit:

Suppose that for a given infinite sequence of points ，，，there is a number l such that every open interval, no matter how small, marked

Off about the point l, contains all the points except for at most a finite number. The number l is then called the limit of the sequence ，，is convergent and converges to l; in symbols, .

The following definition of limit is equivalent:

To any positive number ε,no matter how small ,we can assign a sufficiently large integer N=N（ε）such that from the index N onward [that is, for ngt; N（ε）]we always have .

Of course, it is true as a rule that N（ε）will have to be chosen larger and larger for smaller and smaller values of the toleranceε;in other words, N（ε） will usually increase beyond all bounds as ε tends to zero. The vague intuitive notion of limit suggests a picture of the moving closer and closer to l. This picture is replaced here by the precise “static” definition: Any neighborhood of l contains all with at most a finite number of exceptions.

Obviously, a sequence ，，cannot have more than one limit l. If on the contrary two distinct number l and lrsquo; were limits of the same sequence ，，,we could mark off open interval contains all but a finite number of the ,the sequence could not be infinite. The limit of a convergent sequence is therefore uniquely determined.

Another obvious but useful remark is: If from a convergent sequence we omit any number of terms the resulting sequence converges to the same limit as the original sequence.

A sequence which does not converge is said to be divergent. If as n increase the number increase beyond all positive bounds, we say that the sequence diverges to ;as we have already done occasionally, we write then .Similarly, we write if, as n increases, the numbers increase beyond all bounds in the positive direction. But divergence may manifest itself in other ways, as for the sequence =-1，= 1，=-1，= 1，，whose terms swing back and forth between two different values.

Clearly, neither divergence nor convergence of a sequence is affected by removing finitely many terms.

A sequence ，，is bounded if there is a finite interval containing all points of the sequence. Any finite interval is contained in some finite interval that has the origin as center. Hence the requirement that the sequence is bounded means that there exists a number M such that for all n.

A convergent sequence ，，necessarily is also bounded. For let l be the limit of the sequence. Takingε=1 we find from the definition of convergence that all from a certain N onward lie in the interval of length 2 centered at l. The only terms of the sequence that may lie outside that interval are .We can then, however, find a larger finite interval that also includes .

b. Rational Operations with Limits

From the definition of limit it follows at once that we can perform the elementary operations of addition, multiplication , subtraction, and division of limits according to the following rules.

If ，，is a sequence with the limit a and is a sequence with the limit b, then the sequence of numbers = also has a limit c, and

The sequence of number = likewise converges and

Similarly, the sequence converges and

Provided the limit b differs from zero, the number likewise converge and have the limit

In words, We can interchange the rational operations of calculation wit

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