一类单调递减正项数列的BOX维数估计

Box Dimension Estimationfor a Class of MonotonicallyDecreasing Positive Sequence

当使用比较原则、 柯西判别法在讨论正项级数敛散性时,会遇到比值极限为1从而无法使用的情况,典型的例子为数组{1/n}和{1/n2}的正项级数具有不同的敛散性。 本文讨论了形如1/na(α > 0) 正项级数的敛散性与对应点集的BOX维数存在一定关联,当点集的BOX维数大于等于1/2时,正项级数发散;小于1/2时,正项级数收敛,同时提出了利用1/n为参照对象判断数列正项级数敛散性的一种新方法。

When using the comparison principle and cauchy discriminant method to discuss theconvergence and divergence of positive series, we may encounter situations where theratio limit is 1 and cannot be used. A typical example is that the positive series ofarrays {1/n} and {1/n2} have different convergence and divergence. This article discussesthe convergence and divergence of positive series in the form of 1/na(α > 0), whichis related to the BOX dimension of the corresponding point set. When the BOXdimension of the point set is greater than or equal to 1/2, the positive series diverges;when it is less than 1/2, the positive series converges, and a new method is proposed todetermine the convergence and divergence of the positive series of a sequence using 1/n as the reference object.