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数学与物理
理论数学
Vol. 14 No. 7 (July 2024)
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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
DOI:
10.12677/PM.2024.147293
,
PDF
,
,
,
被引量
下载: 16
浏览: 21
国家自然科学基金支持
作者:
张 云
,
梁永顺
*
:南京理工大学数学与统计学院,江苏 南京
关键词:
正项级数
;
BOX维数
;
敛散性
;
Positive Series
;
BOX Dimension
;
Convergence or Divergence
摘要:
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/n
2
}的正项级数具有不同的敛散性。 本文讨论了形如1/n
a
(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。
Abstract:
When using the comparison principle and cauchy discriminant method to discuss the convergence and divergence of positive series, we may encounter situations where the ratio limit is 1 and cannot be used. A typical example is that the positive series of arrays {1/n} and {1/n
2
} have dierent convergence and divergence. This article discusses the convergence and divergence of positive series in the form of 1/n
a
(α > 0), which is related to the BOX dimension of the corresponding point set. When the BOX dimension 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 to determine the convergence and divergence of the positive series of a sequence using 1/n as the reference object.
文章引用:
张云, 梁永顺. 一类单调递减正项数列的BOX维数估计[J]. 理论数学, 2024, 14(7): 275-283.
https://doi.org/10.12677/PM.2024.147293
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