在数学领域, 收敛性判别法是判断无穷级数收敛, 条件收敛, 绝对收敛, 区间收敛或发散的方法.

如果 r < 1, 那幺级数绝对收敛. 如果 r > 1, 那幺级数发散. 如果 r = 1, 比例判别法失效, 级数可能收敛也可能发散.

其中 "lim sup" 表示上极限 (可能为无穷; 如果极限存在,极限值等于上极限).

如果 r < 1, 级数绝对收敛. 如果 r > 1, 级数发散. 如果 r = 1, 开方判别法无效, 级数可能收敛也可能发散.

那幺级数收敛. 如果积分发散, 那幺级数也发散.

In mathematics, a series is the sum of the terms of an infinite sequence of numbers.

Given an infinite sequence , the nth partial sum is the sum of the first n terms of the sequence, that is,

A series is convergent if the sequence of its partial sums tends to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if there exists a number such that for any arbitrarily small positive number , there is a (sufficiently large) integer such that for all ,