正项级数敛散性的微分判别法

Differential Test for Series of Positive Terms

对正项级数∑ｎｋ＝１ｆ（ｋ），ｆ（ｘ）是相应的正的连续函数，令ｄｄｘ［１ｆ（ｘ）］＝ｇ（ｘ），则ｘ足够大时ｆｇｘ≥１＋α（α＞０）时级数收敛；ｆｇｘ≤１时级数发散。在众多情况下它可以取极限形式。这一微分判别法也是一般函数项级数和无穷限反常积分的判别法。它不仅是简单的，而且是非常普适的。由此讨论了一些例子。

For a series of positive terms ∑nk=1f(k), let f(x) is a corresponding positive continuous function, and ddx1f(x)]=g(x), then the series converges if fgx≥1+α (α>0) for enough big x; the series diverges if fgx≤1. In much cases it may make a limit form. This differential test is also a test of the series of functions and one of the infinite integral. The test is not only simple but very universal and final. From this some examples are discussed.