高數重要極限證明原創中英文對照版

高數重要極限證明原創中英文對照版

重要極限

Important Limit

作者 趙天宇

Author:Panda Zhao

我今天想在這里證明高等數學中的一個重要極限：

Today I want to prove animportant limit of higher mathematics by myself:

想要證明上述極限，我們先要去證明一個數列極限：

If we want to give evidence ofthe limit, first of all, there are a limit of a series of numbers according toa certain rule we need to certify:

想要證明這個極限，我首先要介紹一個定理和一個法則：

Before we begin to prove thelimit, there are one theorem and one rule that are the key point we need to introduce:

1.       牛頓二項式定理（Binomialtheorem）

定理的定義為：

Definition of Binomial theorem:

其中 ，稱為二項式系數，又有 的記法。

Among the formula: we define the  as binomialcoefficient, it can be remembered to.

牛頓二項式定理（Binomial theorem）驗證和推理過程：

The process of the ratiocination of Binomialtheorem:

采用數學歸納法

We consider to use the mathematical inductionto solve this problem.

當n = 1時(While n = 1:),

;

假設二項展開式在n=m時成立。

We can make a hypothesis that the binomial expansionequation is true when n = m.

設n=m+1，則：So if we suppose that n equal mplus one, we will CONTINUE to deduce:
 

具體步驟解釋如下：

The specific step of interpretation :

第三行：將a、b乘入；

The 3rd line: a and b are multiplied into the binomial expansion equation.;

第四行：取出k=0的項；

The 4th line: take out of theitem which includes the k = 0 in the binomial expansion equation.;

第五行：設j=k-1；

The 5th line: making a hypothesisthat is j = k-1;

第六行：取出k=m+1項；

The 6th line: What we need totake out of the item including k=m+1 in the binomial expansion equation.

第七行：兩項合并；

The 7th line: Combining the twobinomial expansion equation.

第八行：套用帕斯卡法則；

The 8th line: At this line weneed to use the Pascal’s Rule to combine the binomial expansion equation whichare
.; 

接下來介紹一下帕斯卡法則(Pascal’s Rule)。

So at this moment, we should get someknowledge about what the Pascal’s Rule is. Let’s see something about it:

帕斯卡法則(Pascal’s Rule)：組合數學中的二項式系數恒等式，對于正整數n、k(k<=n)有：

Pascal’s Rule: a binomial coefficientidentical equation of combinatorial mathematics. For the positive integer n andk (k<=n), there is a conclusion:

                  通常也可以寫成：
                  There is also commonly written:

代數證明：

Algebraic proof:

重寫左邊：

We can rewrite the left combinatorial item:

通分；reductionof fractions to a common.

                                         合并多項式；combining the polynomial.

                          證明完成；The Pascal’s Rule has been proved.

接下來只要要證明是單調增加并且有界的，那么就可以得到它存在極限，我們通常稱它的極限為e。

So what is our next step? The progression ofnumbers according to a certain rule of should be proved that it is a monotonicincrease sequence and has a limitation. If we can do these things, we will drawa conclusion that the sequence has an limitation which we generally call e.

類似的，我們可以得到：

We can analogously get the:

可見，  和相比，除了前兩個1相等之外，后面的項都要小，并且多一個值大于0的項目，因此：

Thus it can be seen, comparing   with  , all of the items of the   are lower thanthese items in  except the 1stand the 2rd one are equaling. In addition it has an item whose value is biggerthan zero that is in the . So we can get a point :

所以數列是單調遞增的得證，接下來證明其有界性：

Because of the point, we can prove thesequence is an monotonic increase sequence, so we remain only one thing shouldbe proved that is the sequence’s limitation. So let’s get it :

可見{  }是有界的，所以根據數列極限存在準則可得：

Thus it can be seen , the sequence of  has a limitation , as we know, we can draw aconclusion by the means of the rule of limitation of sequence exiting: