ความคิดเห็นที่ 33
First of all, I don't quite sure that I understand your question clearly. But first, I have to show my intention for this topic first, so you will understand my purpose. I have to clear myself first becuase this discussion atmosphere seems to converges to Maku atmosphere.
In this topic, I decide to give an easy example to the non-mathematician that for some functions, in particular the function that has finite radius of convergence, the power series can only approximate around the expansion point, not the whole function. To give them an example, I decide to give them the extreme case, which is the function that has zero radius of convergence. The reason I gave this example because it's power series is the zero function which can be easily seen that it will never be the original function. I cannot give the example in comment no. 30 to the beginning of the topic, because non-mathematician cannot see that why the power series cannot represent the original function when x>1.
To me, I want to just give one easy example to let people see. I did not try to ask the question that why this particular function has this behavior. If you read my topic carefully, there is no "question" sentence; there are only "affirmative" sentence.
After I went back and read all your comments again and agian until the last comment, I understand that you think that I am discussing why this function has such a behavior; and you think that my answer is "because Taylor's is local". If my understanding is correct, then you misunderstand me. I have never said that this function has such a behavior because of Taylor's method. No, I haven't.
Say it again, I gave the reader an example to be easily seen that sometimes Taylor approximation is only local. But I didn't say that "This function has this behavior because Taylor's is local". Please understand me.
If you ask me why this function has such a behavior, my answer is the same as yours, essential singularity. It makes this power series has zero radius of convergence.
But this question is not what we and the others discuss at the beginning of the topic. All the comments before your comment no. 21, I try to emphasize everyone that we are considering only real domain, so stop discussing about singularity, we are not trying to answer why this function has this behavior, I just want to give example.
Starting from your comment no. 21, you try to claim that this behavior happens because of essential singularity, which is absolutely correct.
That I said you are "wrong" in comment no. 22, I didn't mean that your explanation why this function has such a behavior because of singularity is wrong, but I did mean that singularity is not what we are discussing here because we only take function in real domain. In the real domain, Taylor's theorem is always true (it's locally approximate, not locally represent; please be careful about these words, they are different!!) and nothing saying about singularity in Taylor's theorem, isn't it?
So your comments after that followed from your misunderstanding to me. All you said in your comments are correct, but they are not what I am discussing in this real domain topic. That's why we are still arguing again and agian. I answer almost everything in real domain and you are talking about complex domain, if this situation continues, we will never meet an agreement and this is bad for other readers because more and more complex topics will be discussed and this easily makes the general reader confused. This is not what I want, as a topic owner, it contradicts with my first intention to give reader some more knowledge in real analysis.
Please ignore my comment no. 30, I misunderstand your question, sorry.
Anything else to discuss? or I am still misunderstand your point? or if you want to discuss complex analysis issues, I will be happy to join you in the newly open topic.
Hope we are on the same page now.
จากคุณ :
CS 
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12 ม.ค. 49 04:01:38
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