No PART of Mathematics suffers more from the triviality of its
initial presentation to beginners than the great subject of series. Two
minor examples of series, namely arithmetic and geometric series, are
considered; these examples are important because they are the simplest
examples of an important general theory. But the general ideas are never
disclosed and thus the examples, which exemplify noting, are reduced to
silly trivialities.
没有哪部分数学比级数这一伟大课题更容易在向初学者介绍时显得琐碎。两个小的级数例子——即算术级数和几何级数——被考虑在内；这些例子之所以重要，是因为它们是一个重要一般理论的最简单例子。但是，普遍的理念从未被揭示出来，因此这些例子——本应展示重要内容——反而沦为无聊的琐事。
The general mathematical idea of a series is that of a set
of things ranged in order, that is, in sequence. This meaning is
accurately represented in the common use of the term. Consider, for
example, the series of English Prime Ministers during the nineteenth
century, arranged in the order of their first tenure of that office
within the century. The series commences with Willian Pitt, and ends
with Lord Rosebery, who, appropriately enough, is the biographer of the
first member. We might have considered other serial orders for the
arrangement of these men; for example, according to their height or
their weight. These other suggested orders strike us as trivial in
connexion with Prime Ministers, and would not naturally occur to the
mind; but abstractly they are just as good orders as any other. When one
order among terms is very much more important or more obvious that other
orders, it is often spoken of as the order of those terms. Thus the
order of the integers would always be taken to mean their order as
arranged in order of magnitude. But of course there is an indefinite
number of other ways of arranging them. When the number of things
considered is finite, the number of ways of arranging them in order is
called the number of their permutations. The number of permutations of a
set of things, where is some finite integer, is
级数的数学一般概念是指一组事物按顺序排列，即按顺序排列的序列。这个意义在该术语的常见用法中得到了准确的表达。例如，考虑19世纪英国首相的序列，按他们在世纪内首次担任该职务的顺序排列。这个序列从威廉·皮特开始，最终由罗斯伯里勋爵结束，恰如其分的是，他是皮特的传记作者。我们本可以考虑其他序列方式来安排这些人物；例如，根据他们的身高或体重来排序。与首相相关的这些其他建议的排序方式在我们看来显得琐碎，而且不太容易想到；但是抽象地说，它们与任何其他排序方式一样合理。当某一排序在各个项之间比其他排序更为重要或更明显时，人们常常将其称为这些项的“排序”。因此，整数的排序通常指它们按大小顺序排列的方式。但当然，还有无数其他方式来排列这些整数。当考虑的事物数量是有限的时，按顺序排列它们的方式的数量被称为它们的排列数。若一个集合包含个事物，其中是某个有限整数，那么这些事物的排列数是
that is to say, it is the product of the first integers; this product is so important
in mathematics that a special symbolism is used for it, and it is always
written ''. Thus, and , and , and . As
increases, the value of increases very quickly; thus is a hundred times as large as .
It is easy to verify in the case of small values of that is the number of ways of arranging
things in order. Thus consider
two things and ; these are capable of the two orders
and , and .
Again, take three things
and ; there are capable of the six
orders,
and . Similarly for the
twenty-for orders in which four things and can be arranged.
When we come to the infinite set of things —— like the sets of all
the integers, or all the fractions, or all the real numbers for
instance—— we come at once upon the complications of the theory of
order-types. This subject was touched upon in Chapter 6 in considering
the possible orders of the integers, and of the fractions, and of the
real numbers. The whole question of order-types forms a comparatively
new branch of mathematics of great importance. We shall not consider it
any further. All the infinite series which we consider now are of the
same order-type as the integers arranged in ascending order of
magnitude, namely, with a first term, and such that each term has a
couple of next-door neighbours, one on either side, with the exception
of the first term which has, of course, only one next-door neighbour.
Thus, if be any integer (not
zero), there will be always an th
term. A sreies with a finite number of terms (say terms) has the same characteristics as
far as next-door neighbours are concerned as an infinite series; it only
differs from infinite series in having a last term, namely, the th.
The important thing to do with a series of numbers——using for the
future 'series' in the restricted sense which has just been
mentioned——is to add its successive terms together.
对一个数列来说——这里将“数列”一词用于刚才提到的狭义的定义——最重要的事情是将其连续的项相加。
Thus if are
respectively the 1st, 2nd, 3rd, 4th,...,nth,...,terms of a series of
numbers, we form successively the series
and so on; thus the sum of the 1st terms may be written
因此，如果分别是数列的第1项、第2项、第3项、第4项，...、第项，...，我们依次形成数列等等；因此，第项的和可以写作 If the series has only a finite number of terms, we come at
last in this way to the sum of the whole series of terms. But, if the
series has an infinite number of terms, this process of successively
forming the sums of the terms never terminates; and in this sense there
is no such thing as the sum of an infinite series.
But why is it important successively to add the terms of a series in
this way? The answer is that we are here symbolizing the fundamental
mental process of approximation. This is a process which has
significance far beyond the regions of mathematics. Our limited
intellects cannot deal with complicated material all at once, and our
method of arrangement is that of approximation. The statesman in framing
his speech puts the dominating issues first and lets the details fall
naturally into their subordinate places. These is, of course, the
converse artistic method of preparing the imagination by the
presentation of subordinate or special details, and then gradually
rising to a crisis. In either way the process is one of gradual
summation of effects; and this is exactly what is done by the successive
summation of the terms of a series. Our ordinary method of stating
numbers is such a process of gradual summation, at least, in the case of
large numbers. Thus 568,213 presents itself to the mind as ——
但为什么要以这种方式依次加总数列的各项呢？答案是，我们在这里象征性地表示了近似的基本心理过程。这是一个具有深远意义的过程，远远超出了数学的范畴。我们的智力有限，无法一次性处理复杂的材料，我们的安排方法就是近似法。政治家在起草演讲时把主要问题放在前面，让细节自然而然地归位。当然，也有相反的艺术方法，即通过呈现从属或特殊的细节来逐步引导想象力，最后达到一个高潮。不管是哪种方式，这个过程都是逐步累积效果的过程；这正是通过依次加总数列的各项所完成的。我们日常表示数字的方法也可以看作是一个逐步累积的过程，至少在处理大数字时是如此。因此，568,213
在脑海中呈现出来的是 —— In the case of decimal fractions this is so more avowedly.
Thus 3.14159 is ——
在十进制小数的情况下，这一点更加明显。因此，是 —— Also, 3 and and and and
are successive approximations to the complete result 3.14159. If we read
568,213 backwards from right to left, starting with the units, we read it in the artistic way,
gradually preparing the mind for the crisis of 500,000.
The ordinary process of numerical multiplication proceeds by means of
the summation of a series. Consider the computation
数字乘法的普通过程是通过加总一个数列来进行的。考虑以下计算 Hence the three lines to be added form a series of which the
first term is the upper line. The series follows the artistic method of
presenting the most important term last, not from any feeling for art,
but because of the convenience gained by keeping a firm hold on the
units' place, thus enabling us to omit some 0's, formally necessary.
But when we approximate by gradually adding the successive terms of
an infinite series, what are we approximating to? The difficulty is that
the series has no 'sum' in the straightforward sense of the word,
because the operation of adding together its terms can never be
completed. The answer is that we are approximating to the of the summation of the series, and
we must now proceed to explain what the 'limit' of a series is.
The summation of a series approximates to a limit when the sum of any
number of its terms, provided the number large enough, is as nearly
equal to the limit as you care to approach. But this description of the
meaning of approximating to a limit evidently will not stand the
vigorous scrutiny of modern mathematics. What is meant by large
enough, and by nearly equal, and by care to
approach? All these vague phases must be explained in terms of the
simple abstract ideas which alone are admitted into pure
mathematics.
Let the successive terms of the series be , so that
is the term of the series. Also let be the sum of the terms, whatever may be. So that ——
让数列的连续项分别为 ，其中 是数列的第 项。又设 为前 项的和，无论 取什么值。所以—— Then the term form a new series,
and the formation of this series is the process of summation of the
original series. Then the 'approximation' of the of the original series to a
'limit' means the 'approximation of the terms of this new series to a
limit'. And we have now to explain what we mean by the approximation to
a limit of the terms of a series.
Now, remembering the definition (given in Chapter 11) of a ,
the idea of a limit means this:
is the limit of the terms of the series if ,
corresponding to each real number
, taken as a standard of approximation, a term of the series can be found so that
all succeeding terms (i.e. ) approximate to
within that standard of
approximation. If another smaller standard be chosen, the term may be too early in the series, and
a later term with the above
property will then be found.
If this property holds, it is evident that as you go along to series
from left
to right, after a time you come to terms which are nearer to
than any number which you may
like to assign. In other words you approximate to as closely as you like. The close
connextion of this definition of the limit of a series with the
definition of a continuous function given in Chapter 11 will be
immediately perceived.
Then coming back to the original series the limit of the
terms of the series is called the
'sum to infinity' of the original series. But it is evident that this
use of the word 'sum' is very artificial, and we must not assume the
analogous properties to those of the ordinary sum of a finite number of
terms without some special investigation.
Let us look at an example of a 'sum to infinity'. Consider the
recurring decimal This
decimal is merely a way of symbolizing the 'sum to infinity' of the
series . The
corresponding series found by summation is
The limit of the terms of this series is ; this is easy to see by
simple division, for
让我们来看一个“无穷和”的例子。考虑循环小数 。这个小数仅仅是表示数列
的“无穷和”。通过求和得到的对应数列是 。这个数列的极限是 ；通过简单的除法就能容易看出这一点，因为
Hence, if is
given (the of the definition),
and all succeeding
terms differ from by
less than ; if is given (another choice
for the of the definition), and all succeeding terms differ
from by less than ; and so on, whatever
choice for be made.
It is evident that nothing that has been said gives the slightest
idea as to how the 'sum to infinity' of a series is intrinsically out of
the question, for the simple reason that such a 'sum', as here defined,
does not always exist. Series which possess a sum to infinity are called
, and those which do not
possess a sum to infinity are called .
An obvious example of a divergent series is the series of
integers in their order of magnitude. For whatever number you try to take as its sum to infinity,
and whatever standard of approximation you choose, by taking enough terms of
the series you can always make their sum differ from by more than . Again, another example of a divergent
series is the
series of which each term is equal to . Then the sum of terms is , and this sum grows without limit as
increases. Again, another example
of a divergent series is the series in
which the terms are alternately
and . The sum of an odd number of
terms is , and of an even number
of terms is . Hence the terms of
the series
do not approximate to a limit, although they do not increase without
limit.
It is tempting to suppose that the condition for to have a sum to
infinity is that should
decrease indefinitely as
increases. Mathematics would be a much easier science than it is, if
this were the case. Unfortunately the supposition is not true.
例如，数列 is divergent. It is easy to see that this is the case; for
consider the sum of terms
beginning at the term.
These terms are :
there are of them and is the least among them.
Hence their sum is greater than
times
is greater than .
Now, without altering the sum to infinity, if it exists, we can add
together neighbouring terms, and obtain the series
是发散的。很容易看出这是正确的；因为考虑从第 项开始的 项的和。这 项是 ：它们共有 项，而
是其中最小的一项。因此，它们的和大于 倍的 ，即大于 。现在，在不改变无穷和的情况下（如果它存在的话），我们可以将相邻的项相加，得到数列
that is , by what has been said above, a series whose terms
after the 2nd are greater than those of the series,
也就是说，根据上述所说，得到的数列，其第二项之后的各项大于原数列的各项。
where all the terms after the first are equal. But this series
is divergent. Hence the original series is divergent.
其中，所有第一项之后的项都相等。但这个数列是发散的。因此，原始数列也是发散的。
The question of divergency shows how careful we must be in arguing
from the properties of the sum of a finite number of terms to that of
the sum of an infinite series. For the most elementary property of a
finite number of terms is that of course they possess a sum: but even
this fundamental property is not necessarily possessed by an infinite
series. This caution merely states that we must not be misled by the
suggestion of the technical term 'sum of an infinite series'. It is
usual to indicate the sum of the infinite series.
发散性的问题表明，我们在从有限项和的性质推论到无限级数和的性质时必须非常小心。因为有限项的最基本性质当然是它们具有和：但即使这个基本性质，无限级数也不一定具备。这个警告仅仅是指出，我们不能被“无穷级数和”这一技术术语的表象所误导。通常会表示无限级数的和。
We now pass on to a generalization of the idea of a series ,
which mathematics, true to its method, makes by use of the variable.
Hitherto, we have only contemplated series in which each definite term
was definite number. But equally well we can generalize, and make each
term to be some mathematical expression containing a variable . Thus we may consider the series and the
series
我们现在转向数列概念的一个概括，数学通过使用变量，忠实于其方法，做出了这一概括。迄今为止，我们只考虑了每一项都是确定数值的数列。但我们同样可以进行概括，使得每一项成为包含变量
的某个数学表达式。因此，我们可以考虑数列 ，以及这个数列 In order to symbolize the general idea of any such function
conceive of a function of
say, which involves in its formation a variable integer , then, by giving the values in succession, we get the
series
为了象征任何此类函数的一般概念，可以设想一个函数 ，其中涉及一个变量整数 ，然后，通过依次赋予 值 ，我们得到数列。 Such a series may be convergent for some values of and divergent for others, It is , in
fact , rather rare to find a series involving a variable which is convergent for all values of
,——at least in any particular
instance it is very unsafe to assume that this is the case. For example,
let us examine the simplest of all instances, namely, the 'geometrical'
series.
这样的数列对于某些 xx
的值可能是收敛的，而对于其他值则是发散的。事实上，找到一个对于所有 xx
值都收敛的数列是相当罕见的——至少在任何特定的实例中，假设这是成立的都是非常不安全的。例如，让我们来研究最简单的一个例子，即“几何”级数。
The sum of terms is
given by Now multiply both sides by and we get Now substract the last line form the upper line and we get
and hence (if be not
equal to ) Now if be numerically
less than 1, for sufficiently large values of , is always numerically
less than ,however be chosen. Thus, if be numerically less than 1, the series
is
convergent, and is
its limit. This statement is symbolized by
But if is numerically
greater than 1, or numerically equal to 1, the series is divergent. In
other words, if lie between and , the series is convergent; but if
be equal to or , or if lie outside the interval to , then the series is divergent. Thus
the series is convergent at all 'points' within the interval to , exclusive of the end points.
At this stage of our inquiry another question arise. Suppose that the
series
在我们探讨的这个阶段，又出现了一个问题。假设级数 is convergent for all values of lying within the interval to , i.e. the series is convergent for any
value of which is greater than
and less than . Also, suppose we want to be sure that
in approximating to the limit we add together enough terms to come
within some standard of approximation . Can we always state some number of
terms, say , such that, if we take
or more terms to form the sum,
then whatever value has
within the interval we have satisfied the desired standard of
approximation?
Sometimes we can and sometimes we cannot do this for each value of
. When we can, the series is
called uniformly convergent throughout the interval, and when we cannot
do so, the series is called non-uniformly convergent throughout the
interval. It makes a great difference to the properties of a series
whether it is or is not uniformly convergent through an interval. Let us
illustrate the matter by the simplest example and the simplest
numbers.
考虑几何级数 It is convergent throughout the interval to , excluding the end values .
But it is not uniformly convergent throughout this interval. For if
be the sum of terms, we have proved that the
difference between and the
limit is . Now suppose be
any given number of terms, say 20, and let be any assigned standard of
approximation, say . Then , by
taking near enough to or near enough to , we can make the numerical value of
to be greater than
. Thus 20 terms will not do
over the whole interval, though it is more than enough over some parts
of it.
The same reasoning can be applied whatever other number we take
instead of 20, and whatever standard of approximation instead of . Hence the geometric series is non-uniformly
convergent over its whole interval of convergence to . But if we take any smaller interval
lying at both ends within the interval to , the geometrical series if uniformly
convergent within it, For example, take the interval to . Then any value for which makes numerically less than
at these limits for also serves for all values of between these limits, since it so
happens that diminishes in numerical value as diminishes in numerical value. For
example, take ; then ,
putting we
find:
相同的推理可以应用于我们取任何其他数字代替20，并且代替的近似标准也是如此。因此，几何级数
在其整个收敛区间 到
上是非均匀收敛的。但是，如果我们取一个位于区间 到
内部的较小区间，那么几何级数在该区间内是均匀收敛的。例如，取区间 到 。然后，对于任何使得
在这些区间的端点处的数值小于的，它同样适用于这些端点之间的所有值，因为恰好是 随着数值的减小而减小。例如，取；然后，代入，我们得到： Thus three terms will do for the whole interval, though, of
course, for some parts of the interval it is more than is necessary.
Notice that, because
因此，三个项就足够覆盖整个区间，尽管当然在区间的某些部分，它的项数多于所需的数量。注意，由于
is convergent (though not uniformly) throughout the interval to , for each value of in the interval some number of terms
can be found which will satisfy a
desired standard of approximation; but, as we take nearer and nearer to either end value
or , larger and larger values of have to be employed.
It is curious that this important distinction between uniform and
non-uniform convergence was not published till 1847 by
Stokes——afterwards, Sir George Stokes——and later, independently in 1850
by Seidel, a German mathematician.
The critical points, where non-uniform convergence comes in, are not
necessarily at the limits of the interval throughout which convergence
holds. This is a speciality belonging to the geometrical series.
非均匀收敛出现的临界点不一定位于收敛区间的端点。这是几何级数的一个特殊性质。
In the case of the geometric series , a simple
algebraic expression
can be given for its limit in its interval of convergence. But this is
not always the case. Often we can prove a series to be convergent within
a certain interval, though we know nothing more about its limit except
that it is the limit of the series. But this is a very good way of
defining a function; viz. as the limit of an infinite convergent series,
and is , in fact, the way in which most functions are, or ought to be,
defined.
Thus, the most important series in elementary analysis is
因此，在初等分析中，最重要的级数是
Where has the meaning defined
earlier in this chapter. This series can be proved to be absolutely
convergent for all values of , and to be uniformly convergent within
any interval which we like to take. Hence it has all the comfortable
mathematical properties which a series should have. It is called the
exponential series. Denote its sum to infinity by exp. Thus, by definition,
It is fairly easy to prove, with a little knowledge of elementary
mathematics, that
用一些初等数学知识，证明这一点是相当简单的，具体来说，
in other words that This property is an
example of what is called an addition-theorem. When any function [say
] has been defined, the first
thing we do is to try to express in terms of known functions of
only, and know functions of only. If we can do so, the result is
called an addition-theorem. Addition-theorems play a great part in
mathematical analysis. Thus the addition-theorem for the sine is given
by
这个性质
是所谓的加法定理的一个例子。当一个函数（比如 ）已经被定义时，我们做的第一件事就是尝试将
用仅与 相关的已知函数和仅与
相关的已知函数表示。如果我们能够做到这一点，那么这个结果就被称为加法定理。加法定理在数学分析中起着重要作用。比如，正弦函数的加法定理是
and for cosine by
As a matter of fact the best ways of defining sin and cos are not by the elaborate geometrical
methods of the previous chapter, but as the limits respectively of the
series
These definitions are equivalent to the geometrical definitions, and
both series can be proved to be convergent for all values of , and uniformly convergent throughout
any interval. These series for sine and cosine have a general likeness
to the exponential series given above. They are, indeed, intimately
connected with it by means of the theory of imaginary numbers explained
in Chapter 7 and 8.
The graph of the exponential function is given in Fig.29. It cuts the
axis at the point , as evidently it ought to do , since
when every term of the series
except the first is zero. The importance of the exponential function is
that it represents any changing physical quantity whose rate of increase
at any instant is a uniform percentage of its value at that instant. For
example, the above graph represents the size at any time of a population
with a uniform birth-rate, a uniform death-rate, and no emigration,
where the corresponds to the time
reckoned from any convenient day, and the represents the population to the proper
scale. The scale must be such that represents the population at the date
which is taken as the origin. But we have here come upon the idea of
'rates of increase' which is the topic for the next chapter.
An important function nearly allied to the exponential function is
found by putting—— for as the argument in the exponential
function. We thus get .
the graph is given in
Fig.30.
The curve, which is something like a cocked hat, is called the curve
of normal error. Its corresponding function is vitally important to the
theory of statistics, and tells us in many cases the sort of deviations
from the average results which we are to expect.
Another important function is found by combining the exponential
function with the sine, in this way:
另一个重要的函数是通过将指数函数与正弦函数结合得到的，具体方式如下：
Its graph is given in Fig.31. The point are placed at equal
intervals , and an
unending series of them should be drawn forwards and backwards. This
function represents the dying away of vibrations under the influence of
friction or of 'damping' forces. Apart from the friction, the vibrations
would be periodic, with a period ,
but the influence of the friction makes the extent of each vibration
smaller than that of the preceding by a constant percentage of that
extent. This combination of the idea of 'periodicity' (which requires
the since or cosine for its symbolism) and of 'constant percentage'
(which requires the exponential function for its symbolism) is the
reason for the form of this function, namely, its form as a product of a
sine-function into an exponential function.
if a series with all its terms
positive is convergent, the modified series found by making some terms
positive and some negative according to any definite rule is also
convergent. Each one of the set of series thus found, including the
original series, is called 'absolutely convergent'. But it is possible
for a series with terms partly positive and partly negative to be
convergent, although the corresponding series with all its terms
positive is divergent. For example, the series
如果一个所有项均为正数的数列是收敛的，那么通过按照某个明确规则将一些项变为正数、一些项变为负数所得到的修改后的数列也是收敛的。这样得到的每一个数列集合，包括原始数列，都被称为“绝对收敛”的数列。但是，对于一个部分为正、部分为负的数列，它可能是收敛的，尽管将其所有项变为正数后得到的数列是发散的。例如，数列
is convergent though we have just proved that
是收敛的，尽管我们刚刚证明了 is divergent. Such convergent series, which are not absolutely
convergent, are much more difficult to deal with than absolutely
convergent series.
