正项级数敛散性判别法的源与流

Extension of the test of convergence and divergence of series of positive terms

利用正项级数的比较判别法这个源头,通过不同的后台级数尝试着揭示许多判别法的发现过程,从中发现了一种普遍的方法和规律,即利用标准级数的适当组合及其参数判别敛散性,再用一般级数代替加以验证,并将这种规律进行拓展与创新获得2种新的判别法,即若正项级数∑∞n=1un,有limn→∞lnnln lnnnlnnn1un-1=pnl→im∞lnnnnu1n-1=p。当p>1时,∑n∞=1un收敛,当p<1时,∑n∞=1un发散。

Based on the ratio test of series of positi various test methods in terms of different backgroun ve terms and the search of the forming process of d series, a generalized rule of testing the convergence and divergence by proper combination of the standard series and their parameters and testifying by substitution with normal series is established. Two new test methods are proposed via extension of the rule. In the series of positive terms ∑n=1^∞un,that satisfies lim n→∞ln n/ln ln n[n/ln n（n√1/um-1））]=plimn→∞n/ln n（n√1/un-1）=p.if p〉1时,∑n=1^∞un converges,vut if P〈1时,∑n=1^∞um that satisfies