无穷级数

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无穷级数就是可数的无穷多个数相加,其和定义为前n项之和当n趋向于无穷大时的极限;如果极限不存在,就说级数是发散的。

在无穷级数的理论中,有三件事要做:

1)收敛性的判定。除了数列极限的存在性准则外,数学家们还发展了许多充分性的判别法;诸如正项级数的比较判别法、比值判别法、根值判别法、积分判别法等等;交错级数的迪里克莱判别法,一般变号级数的绝对收敛性判别法、用和差变换进行的Abel判别法。最关键的概念是,每一项必须小到一定程度,级数才可能收敛。

2)级数的运算与求和。我们可以定义级数的加、减法,在收敛的意义下,其和也相等;但对于乘法,可以有不同的组合方式(通常是按对角线方式相加),由于级数求和不能任意交换顺序,只有在某个级数绝对收敛时,其乘积才会等值。级数的除法就很难进行了,只有幂级数或者按照某种特数函数列展开的级数,才有可能用乘法的逆运算去做除法。

对于函数项级数,若要进行求导或积分等分析运算,通常要求一致收敛性才行。在这方面,幂级数和三角级数具有很好的性质,因而可以广泛地用于函数的表示。

级数的求和是一个难题。通常的办法有拆项法(适用范围有限)、幂级数求和法、三角级数求和法。欧拉求和公式可以把某些级数变换为积分,而分部求和法(和差变换法)可以把一个级数变成另一个级数,其收敛性或许加快或许放慢。在函数项级数中,也可以进行变量代换,只要保证收敛性就行。

3)函数的无穷级数表示。我们可以按照任何一列函数(幂函数、正交函数等)来展开一个给定的函数,只要运算方便即可。最常用的是幂级数,我们有各个基本初等函数的泰勒展开式;对于周期函数,则用三角级数(又叫傅立叶Fourier级数),尽管求导数不方便,积分却是随便可以进行的。

 

我们还可以用无穷乘积来表示函数,这相当于对函数进行因式分解,必须求出它的所有零点才行。尽管有复变函数的Hadamard定理,我们所知的可以用无穷乘积表示的函数屈指可数,大概不超过十个。

人们对无穷大的理解还十分有限。由于无法把无穷大当作一个数,对于发散级数,人们无能为力。正如Stephen Hawking所说的,数学对于无穷大的数无能为力。数学家们就此认输吗?

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