正项级数微分判别法及其完备性和非标准分析

New Differential Test for A Series of Positive Terms and Its Completeness and Nonstandard Analysis

证明了正项级数的一种新微分判别法:∑∞k=1f(k)是正项级数,令f(x)是相应的正连续函数,且ddx[f(1x)]=g(x),如果f(x)g(x)x≥1+α(α>0),级数收敛;如果f(x)g(x)x≤1,级数发散.这一判别法简单易推广,结合非标准分析,论述了微分判别法的完备性,同时该方法也是一般的函数项级数和无穷限积分敛散性的判别法.

A new differential test for series of positive terms is proved.Let ^∞∑k=1 f（k） be a series of positive terms,f（x） is a corresponding positive continuous function,and d/dx[1/f（x）]=g（x）.Then,if f（x）g（x）x≥1＋α（α〉0）,the series converges;if f（x）g（x）x≤1,the series diverges.This test is simple,and can be applied widely.Combining the nonstandard analysis,the completeness of the differential test is discussed.It is also the test of the general series of functions and the infinite integral,whose convergence or divergence may be determined by the differential of the functions.