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А.Я辛钦在“教学分析简明教程”§78函数的冪级数展开式一节中指出,只有当函数S(x)在给定的区间的每一点处存在任意阶的导数时,才能谈到这个函数展开成冪级数的问题。如果函数S(x)可以展成冪级数 S(x)=sum from n=0 to ∞(a_nx~n), (1)则这个级数就一定具有所谓焉克洛林级数的形式 S(x)=sum from n=0 to ∞((S~(n))(0))/(n!)x~n (2) 辛钦指出,任何一个在x=0处具有任意阶的导数的函数,都有焉克洛林级数(2);当然,这还并没有能解决掉这些关于把函数S(x)展开成冪级数的问题,因为:1)级数(2)在任何一点x≠0处都可能是发散的;2)即使级数(2)在点x≠0处收敛,它的和也还可能不等于S(x)。
In the section “Power series expansion” of the §78 function of the “Concise Tutorial for Teaching Analysis” section, A. R. Sinqin pointed out that only when the function S(x) exists an arbitrary order derivative at each point of a given interval, Speaking of the problem that this function expands into a power series. If the function S(x) can be developed into a power series S(x) = sum from n = 0 to ∞(a_nx~n), (1) then this series must have the form of the so-called Crohn’s series S (x)=sum from n=0 to ∞((S~(n))(0))/(n!)x~n (2) Xin Qin points out that any derivative with an arbitrary order at x=0 The function has a 焉Crolean series (2); of course, this doesn’t solve the problem of expanding the function S(x) into a power series because: 1) the number of stages (2) is Any point x≠0 may be divergent; 2) Even if the series (2) converges at point x≠0, its sum may not be equal to S(x).