Abstract: 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.

Key words: positive terms, rest, differential, nonstandard analysis