无穷小在极限及正项级数方面的应用

Application of Infinitesimal in Limit and Positive Series

以无穷小商的极限等价代换为基础,推广并论述了等价无穷小在和差极限运算中的运用;指出了两同阶无穷小的和差运算在满足何种条件下可逐项等价代换,两不同阶的无穷小在满足何种条件下其高阶无穷小可以略去,使等价无穷小替换由积商型结构推广到和差型的结构中;这样可以大大简化极限的计算过程,能清楚在和差极限运算时,什么时候可逐项代换,什么时候可以略去;并举例说明了二个定理在极限计算中的应用.同时用无穷小的比较观点来解释正项级数敛散性判别的极限形式,对于理解和使用该判别法有大的帮助,它也是无穷小的一个应用.

The limit operation on sum-difference with infinitesimal, based on equivalent infinitesimal replacement, is discussed. And sufficient conditions for equivalent replacement in the limit operation on sum-difference with infinitesimal of same or not same order are given. The structure of product-quotient with equivalent infinitesimal replacement is extended to sum-difference, thus calculation process of limit may be greatly simplified, enabling us aware of when to use equivalent replacement and when to omit it while calculating sum-difference limit. And application of the two theorems in limit calculation. This is illustrated with examples. Meanwhile, the limit form of discriminance about convergence and divergence of positive series compared with infinitesimal are explained. Its application of infinitesimal is explained, and it is of much help for us to understand the discriminance clearly.