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The original Chinese and English version of the important limit proof of high numbers

Important limit

Important Limit

Author Zhao Tianyu

Author:Panda Zhao

I want to prove an important limit in higher mathematics here today:

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

To prove the above limit, we first have to prove a sequence limit:

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:

To prove this limit, I first introduce a theorem and a rule:

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

1. Newton binomial theorem (Binomialtheorem)

The definition of the theorem is:

Definition of Binomial theorem:

Among them, it is called binomial coefficient, and there is some notation.

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

The verification and reasoning process of Newton binomial theorem (Binomial theorem):

The process of the ratiocination of Binomialtheorem:

Adopt mathematical induction method

We consider to use the mathematical inductionto solve this problem.

When n = 1 (While n = 1:)

Suppose the binomial expansion holds when n = m.

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

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

The specific steps are explained as follows:

The specific step of interpretation:

The third line: multiply an and b into

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

Line 4: take out the item of kryp0

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

Line 5: set j=k-1

The 5th line: making a hypothesisthat is j = kMui 1

Line 6: take out the k=m+1 item

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

Line 7: merge the two items

The 7th line: Combining the twobinomial expansion equation.

Line 8: apply Pascal's rule

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

.

Next, let's introduce the Pascal rule (Pascal's Rule).

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

Pascal's Rule: binomial coefficient identities in combinatorial mathematics, for positive integers n, k (k)

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