数学专业英语(吴炯圻)(7)

11

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是当n无限增加时，项 f(n)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。做法如下：

12

DIFINITION. A sequence {f(n)} is said to have a limit L if, for every positive number e, there is another positive number N ( which may depend on e ) such that

| an－L | < 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。

13

In this definition the function values f(n) and the limit L may be 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f(n) 和极限L都可能是实数或者复数。如果f和L是复数, 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 the real part u and the imaginary part v converge separately. (P61 第三段最后一句)

换句话说，复值序列f收敛当且仅当实部和虚部(序列)分别收敛。

14

※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, 每一个对所有正实数 x 有定义的函数都可以用来构造一个序列，其办法是限制 x 只取整数值。

15

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 course, divergent sequences that do not have infinite limits.

仅当一个序列的极限有限时，才使用短语“收敛序列”。一个具有无穷极限的序列称为发散