数学专业英语X资料
合集下载
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
formulating these theorems 公式化这些定理
be similar to 和什么相似
作为讨论和、积等的极限的基本规则对于收敛序列的极限 也是成立的。读者自己公式化这些定理应该不会有困难， 它们的证明有点类似于3.5节中给出的那些证明.
divergent sequence 发散序列
术语“收敛序列”通常仅指极限为有限的序列，具有无限 极限的序列称为发散。当然存在没有无限极限的发散序列 ，有下列公式定义的序列，就是例子f(n)=(1)ⁿ,f(n)=sin(nￗ/2),f(n)=(-1)ⁿ(1+1/n),f(n)=e(nￗi/2).
The basic rules for dealing with limits of sums ,products ,etc., also hold for limits of convergent sequence.The reader should have no difficulty in formulating these theorems for himself. Their proofs are somewhat similar to those given in Section 3.5.
show that the relation f(n)→L implies v(n)→a
and u(n)→b as n→∝.Conversely ,the inequality
|f(n)-L|≦|u(n)-a|+|v(n)-b| shows that the two
relations u(n)→a and v(n)→b imply f(n)→L as
数学专业英语
信息151 肖猛 王祎
二：The limit of a sequence
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 increase infinitely.To treat this problem, we must extend the limit concept to sequence.This is done as follows.
It is clear that 显然 限制
restrict
to draw an analogy between 把某事物和某事物做类比
显然，对所有正实数x有定义的函数都可以通过限制x仅 取正整数来构造序列，这表明，刚刚给出的定义与6.4节 中作为更一般函数的定义之间十分类似
The analogy carries over to infinite limits as well ,and we leave it for the reader to define the symbols lim(n→∝)f(n)=+∝ and lim(n→∝)f(n)=-∝ as was done in section 6.5 when f is real-valued. If f is complex, we write f(n)→∝ as n→∝ if |f(n)|→+∝.
we
have
lim(n→∝)f(n)=lim(n→∝)u(n)+i*lim(n→∝)v(n).
conversely 相反的 sequence复值序列
complex-valued
表明当n→∝时，关系式f(n)→L推得v(n)→a和u(n)→b 。反过来，不等式|f(n)-L|≦|u(n)-a|+|v(n)-b|表明当 n→∝时，u(n)→a和v(n)→b，推得f(n)→L。换句话说， 复值序列f收敛当且仅当实部u和虚部v分别收敛，这是有 lim(n→∝)f(n)=lim(n→∝)u(n)+i*lim(n→∝)v(n)
decompose 分解，腐烂 成某物
to decompose into 分解
在这个定义中，函数值f(n)和极限L可以是实数或者复数 。如果f(n)和L是复数，我们可将它们分解成实部和虚部 ，记为f=u+iv, L=a+ib, 则有f(n)-L= u(n)- a+i[v(n)b]. 不等式|u(n)-a|≦|f(n)-L| and |v(n)-b|≦|f(n)L|
limit L if, for every positive number ε, there is another positive number N ( which may depend on ε ) such that | an－L |<ε for all n ≥ N . In this case, we say the sequence {f(n)} converges to L and we write.lim(n→∝)f(n)=L, or f(n)→L as n→∝.A sequence which does not converge is called divergent.
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.This explains the strong analogy between the definition just given and the one in Section 6.4 for more general functions.
n→∝.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 ,in which
case
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 sequence that do not have
infinite limits.Examples are defined by the
following
formulas:f(n)=(-
1)ⁿ,f(n)=sin(nￗ/2),f(n)=(-1)ⁿ(1+1/n),f(n)=e(nￗi/2).
convergent formulas 例子
sequence
收敛序列
converge 汇聚
divergent 有分歧的，发散
Байду номын сангаас
定义：{F(n)}序列据说有限制L如果对于每一个积极的 数字e，有另一个积极的号码N（这可能取决于电子）例如 | an－L | < e for all n ≥ N ，这种情况下，我们说 的序列{f(n)}汇聚为L和我们写lim(n→∝)f(n)=L, or f(n)→L as n→∝不衔接的一系列被称为发散。
sequence 一系列 in sequence of 按某种次序
finite 有限的 合/有限数
a finite set/number有限集
我们担心在这里主要的问题在于决定是否条款f(n)倾向 于有限的n无限增加，若要把这个问题，我们必须扩展序列 的极限概念。这样做，如下图所示。
DEFINITION: A sequence {f(n)} is said to have a
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. Then we have f(n)-L=u(n)-a+i[v(n)-b].The inequalities |u(n)-a|≦|f(n)-L| and |v(n)b|≦|f(n)-L|
carry
over
to
infinite limits 无穷极限
继续做下去
这种类似也可以推广到无穷极限，我们把定义记号 lim(n→∝)f(n)=+∝ and lim(n→∝)f(n)=-∝留给读者 ，就像在6.5节当f是实值时那样做，如果f是复的,当 n→∝是，若|f(n)|→+∝,就记f(n)→∝
be similar to 和什么相似
作为讨论和、积等的极限的基本规则对于收敛序列的极限 也是成立的。读者自己公式化这些定理应该不会有困难， 它们的证明有点类似于3.5节中给出的那些证明.
divergent sequence 发散序列
术语“收敛序列”通常仅指极限为有限的序列，具有无限 极限的序列称为发散。当然存在没有无限极限的发散序列 ，有下列公式定义的序列，就是例子f(n)=(1)ⁿ,f(n)=sin(nￗ/2),f(n)=(-1)ⁿ(1+1/n),f(n)=e(nￗi/2).
The basic rules for dealing with limits of sums ,products ,etc., also hold for limits of convergent sequence.The reader should have no difficulty in formulating these theorems for himself. Their proofs are somewhat similar to those given in Section 3.5.
show that the relation f(n)→L implies v(n)→a
and u(n)→b as n→∝.Conversely ,the inequality
|f(n)-L|≦|u(n)-a|+|v(n)-b| shows that the two
relations u(n)→a and v(n)→b imply f(n)→L as
数学专业英语
信息151 肖猛 王祎
二：The limit of a sequence
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 increase infinitely.To treat this problem, we must extend the limit concept to sequence.This is done as follows.
It is clear that 显然 限制
restrict
to draw an analogy between 把某事物和某事物做类比
显然，对所有正实数x有定义的函数都可以通过限制x仅 取正整数来构造序列，这表明，刚刚给出的定义与6.4节 中作为更一般函数的定义之间十分类似
The analogy carries over to infinite limits as well ,and we leave it for the reader to define the symbols lim(n→∝)f(n)=+∝ and lim(n→∝)f(n)=-∝ as was done in section 6.5 when f is real-valued. If f is complex, we write f(n)→∝ as n→∝ if |f(n)|→+∝.
we
have
lim(n→∝)f(n)=lim(n→∝)u(n)+i*lim(n→∝)v(n).
conversely 相反的 sequence复值序列
complex-valued
表明当n→∝时，关系式f(n)→L推得v(n)→a和u(n)→b 。反过来，不等式|f(n)-L|≦|u(n)-a|+|v(n)-b|表明当 n→∝时，u(n)→a和v(n)→b，推得f(n)→L。换句话说， 复值序列f收敛当且仅当实部u和虚部v分别收敛，这是有 lim(n→∝)f(n)=lim(n→∝)u(n)+i*lim(n→∝)v(n)
decompose 分解，腐烂 成某物
to decompose into 分解
在这个定义中，函数值f(n)和极限L可以是实数或者复数 。如果f(n)和L是复数，我们可将它们分解成实部和虚部 ，记为f=u+iv, L=a+ib, 则有f(n)-L= u(n)- a+i[v(n)b]. 不等式|u(n)-a|≦|f(n)-L| and |v(n)-b|≦|f(n)L|
limit L if, for every positive number ε, there is another positive number N ( which may depend on ε ) such that | an－L |<ε for all n ≥ N . In this case, we say the sequence {f(n)} converges to L and we write.lim(n→∝)f(n)=L, or f(n)→L as n→∝.A sequence which does not converge is called divergent.
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.This explains the strong analogy between the definition just given and the one in Section 6.4 for more general functions.
n→∝.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 ,in which
case
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 sequence that do not have
infinite limits.Examples are defined by the
following
formulas:f(n)=(-
1)ⁿ,f(n)=sin(nￗ/2),f(n)=(-1)ⁿ(1+1/n),f(n)=e(nￗi/2).
convergent formulas 例子
sequence
收敛序列
converge 汇聚
divergent 有分歧的，发散
Байду номын сангаас
定义：{F(n)}序列据说有限制L如果对于每一个积极的 数字e，有另一个积极的号码N（这可能取决于电子）例如 | an－L | < e for all n ≥ N ，这种情况下，我们说 的序列{f(n)}汇聚为L和我们写lim(n→∝)f(n)=L, or f(n)→L as n→∝不衔接的一系列被称为发散。
sequence 一系列 in sequence of 按某种次序
finite 有限的 合/有限数
a finite set/number有限集
我们担心在这里主要的问题在于决定是否条款f(n)倾向 于有限的n无限增加，若要把这个问题，我们必须扩展序列 的极限概念。这样做，如下图所示。
DEFINITION: A sequence {f(n)} is said to have a
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. Then we have f(n)-L=u(n)-a+i[v(n)-b].The inequalities |u(n)-a|≦|f(n)-L| and |v(n)b|≦|f(n)-L|
carry
over
to
infinite limits 无穷极限
继续做下去
这种类似也可以推广到无穷极限，我们把定义记号 lim(n→∝)f(n)=+∝ and lim(n→∝)f(n)=-∝留给读者 ，就像在6.5节当f是实值时那样做，如果f是复的,当 n→∝是，若|f(n)|→+∝,就记f(n)→∝