How to better understand and memorize Taylor expansion

The core idea of this paragraph is imitation. When we want to imitate something, we will follow the idea mentioned above invisibly, that is, first ensure that it is roughly similar, then ensure partial similarity, then ensure that the details are similar, and then ensure that the more subtle places are similar... Constantly After the refinement continues, after infinite refinement, the counterfeit will be infinitely close to the real product. The truth is hard to tell. This is a life experience that everyone understands.

When a physicist applies this life experience to his own research, the following scenario will appear: a car traveling at will, out of a very strange trajectory curve: the
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physicist finds this trajectory very interesting , I also want to drive down a track that is exactly the same. Since it is a copy, he applied the life experience of “fake” to this, and proposed a solution: Since you want to imitate the car just now, you should first ensure that the initial position is the same, and continue to imitate, so that the speed of the car in the initial position is also The same, if not satisfied, continue to refine, this time keep the position and the speed at the initial position the same, while ensuring that the acceleration of the car at the initial position is also the same, if not satisfied, continue to refine, this time ensure the initial position and initial position The speed at the initial position and the acceleration at the initial position are the same, and the rate of change of the acceleration at the initial position is also the same. The physicist came to the conclusion: apply the experience of “fake” in life to kinematic problems. If you want to imitate a curve, you should first ensure that the starting point of the curve is the same, and secondly, ensure the rate of change of the displacement with time at the starting point. Same (the same speed), once again, it should be ensured that the first two are equal and the second-order rate of change with respect to time is the same (the same acceleration)... If the rate of change of each order (each derivative) is the same over time, then these two curves It must be completely equivalent.

1. Taylor

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The following is a rigorous calculation.

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Let's count it as a first-order one.

Here is another second-order one.

At this point, not only Taylor, we ordinary people can also roughly imagine that if we continue to increase the order, the similarity range will continue to expand, and after infinite high order, the entire curve will be infinitely similar. Insert a picture and use a computer to quickly realize it.
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Taylor's story is over, but things are not over, because Taylor didn't tell you how many times to seek guidance .

So, a bunch of people were left to help him wipe his ass. The first one to wipe his ass was Peano. He rounded out the ellipsis in the above formula. However, it eventually ran aground and was not very useful. Later, Lagrange jumped out again to help Peiano wipe his ass. So far the story ends.

Let me talk about Peano's story first.

2. Peano

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Peano began to think about the error. Leaving Peiano aside, if you were to think about this question, what kind of thinking would you have? Since it is an error, it must be as small as possible, right. So when we think about errors, the natural logic is to make this error approach zero. Peano thinks the same way. His general direction is to make the latter half approximately equal to 0.

Once the second half is very close to 0, it can be omitted, and only expand to nth order. Taylor expansion can be used. But he didn't know how to do it.

Later, he began to ponder Taylor’s whole idea: first make sure that the initial point positions are the same, then make sure that the first derivative is the same, which is a bit similar, and then make sure that the second derivative is the same, more refined, and then make sure that the third derivative is the same... suddenly A flash of light: **Taylor expansion is a process of gradual refinement, that is, each item is more refined (smaller) than the previous one. **For example, you want to add 90 catties of grain to 100 catties. The first time you add a large amount, it becomes 99 catties. The second time, you add a small amount and it becomes 99.9 catties. Three times, I added a handful and turned into 99.99 catties...Each time I caught less grain than the previous one. The same is true in Taylor's expansion:
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It can be seen that the last term (nth order) is the smallest. Piano thought: As long as the total error (the sum of all the following items) is smaller than this one, can't the error be ignored?

The task now is to compare the size, compare the size of the last term in the Taylor expansion with the size of the error term, namely:
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How to compare the size? High school students know that comparing big and small is nothing more than doing errands or doing business. If you are not sure, try them one by one. In the end, Piano used the quotient. He divided the error term by the smallest term in Taylor's expansion, and got it after sorting it out:
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I don’t know how you feel when you see it here. Maybe you think Peano is great, or you may think that this is not fooling people.

Anyway, in order to commemorate Peano's contribution, everyone puts the above error term as Peano's residual term.

Summarize Peano’s thoughts: First, he completes those terms that are not written in the Taylor expansion, and then he calls the sum of these terms the error term. After that, he wants to change the error term to 0, and consider When the term in the Taylor expansion is getting smaller and smaller, he divides the error term by the last term, trying to get a result of 0, and finally finds that only when $x$ approaches $x_0$ At that time, the quotient approaches zero, and that's it.

Peano’s story is over. He wanted to perfect Taylor's development, but his results can only be counted as $x$ approaches $x_0$ Time. At this time, Lagrang appeared on the scene.

3. Lagrange

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To describe the simple problem above in mathematical language is Lagrange's median theorem:

Later, Lagrange's median theorem was seen by Cauchy. Cauchy is so awesome, he is naturally sensitive to calculations. Cauchy believed that the ordinate is a function of the abscissa, so I can also write the abscissa as a function, so he proposed the Cauchy median theorem:
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Lagrange was indignant when he heard about this, and felt very sorry. Obviously it was his own way of thinking. Just one step away, Cauchy was able to take advantage of it, but Cauchy did make sense.

This incident left a deep psychological shadow on Lagrange. Next, Lagrange began to think about the error of Taylor series. Like Peano, he only considered the error part (see above).

Add a word, old iron, the next Lagrangian operation is completely broken, I really can't make up his brain circuit.

First of all, like Peano, write the error term first, and set the error term as $R (x)$ :
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Lagrange continues to copy this idea, and wants to see if he can continue to write:

Look
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at the numerator first and then the denominator.

This article covers Taylor expansion, Peano's remainder, Lagrange's median theorem, Cauchy's median theorem, and Lagrange's remainder. The full text is over.

Author: "logged out"
link: https: //www.zhihu.com/question/25627482/answer/313088784
Source: know almost
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4. McLaughlin series

The Taylor series obtained by the derivative of the function at the zero point of the independent variable is also called the McLaughlin series, named after the Scottish mathematician Colin McLaughlin.
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