级数与无穷积分余项收敛性的应用

Applications of Convergence of Remainder of Series and Infinite Integrals

利用级数和无穷积分与其余项的敛散性完全相同这一基本事实,研究了级数和无穷积分的敛散性,由于级数和无穷积分从某个充分大的项开始以后一般具有某种一致性,因此余项的敛散性往往更易于判定。采用级数的余项研究了一个与对数有关的级数的敛散性,并将指数和底数中对数的重数推广到了有限的情形,给出了其敛散性的判定。利用无穷积分的余项证明了两个有关无穷积分收敛结果的推广,讨论了在无穷积分收敛的条件下,被积函数在无穷远处必趋于零的一些充分条件。

A basic fact is applied to investigate the convergence of series and infinite integrals that series and infinite integrals have completely the same comvergence with its remainders.Since series and infinite integrals possess consistency after a certain large term,usually it is easy to verify the convergence of its remainders.The convergence of a series related to natural logarithm is studied by its remainders,and it is generalized to the cases of multi-logarithm for its power and base.Two generalizations of convergence of infinite integrals are proved by its remainders.For convergent infinite integrals,some sufficient conditions are dicussed that guarantee the integrands tending to zero at infinite.