再论无穷多个无穷小量的乘积

对文6]提出的质疑给出回答,表明由于不同的无穷小量趋近于0的速度有快有慢,因此无穷多个无穷小量的乘积∏∞k=1{x_n~(k)}∞n=1,有可能不是无穷小量(其中对每个正整数k,{x_n~(k)}_(n=1)~∞表示极限为0的数列),而验证∏∞k=1{x_n~(k)}∞n=1是否是无穷多个无穷小量的乘积,只需验证对每个正整数k,当n→+∞时,{x_n~(k))_(n=1)~∞是否趋近于0,而无需考虑函数列{{x_n~(k)}_(n=1)~∞}_(k=1)~∞的极限limk→∞x_n~(k)是不是无穷小量.进而,对无穷多个无穷小量的乘积是无穷小量或不是无穷小量给出了一些充分条件,

An Further Notations on the Product of Infinite Infinitesimals

An positive answer is given to the question which is put forward in document 6].It is shown that the product of infinite infinitesimals∞Πk=1{x(k)n}∞n=1may not be infinitesimal because the speeds of the infinitesimals converging to zero are different, where for each positive integer k,｛x(k)n｝∞n=1 is used to denote the series whose limit is zero.To verify that ∞Πk=1{x(k)n}∞n=1 is the product of infinite infinitesimals, we only show that limn→∞｛x(k)n｝∞n=1=0 for each positive integer k and it is not necessary to consider whether the limit of the series {｛x(k)n｝∞n=1}∞k=1 is infinitesimal or not.Moreover, the sufficient conditions under which the product of infinite infinitesimals is infinitesimal or not are obtained.