柯西积的收敛性

IF 0.4 4区数学 Q4 MATHEMATICS

American Mathematical Monthly Pub Date : 2023-05-18 DOI:10.1080/00029890.2023.2206328

Adam Krupowies, F. Prus-Wiśniowski

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引用次数: 0

摘要

一般情况下，绝对收敛级数与条件收敛级数的柯西积是绝对收敛的。在我们的笔记中，我们提供了一种简单而一般的方法来构造这样的级数对，这种方法与经典的Pringsheim例子无关。此外，我们还观察到，当只考虑两个都满足交替级数检验假设的交替级数对时，如果其中一个是绝对收敛的，则它们的柯西积的收敛性质与第二个因子的收敛性质完全相同。我们用柯西积上的沃斯定理的一个新的、惊人的简短证明来完成这些评论。

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The Character of Convergence of the Cauchy Product

Abstract In general, the Cauchy product of an absolutely convergent series and a conditionally convergent one might converge absolutely. In our note, we provide an easy and quite general method for construction of such pairs of series, a method that is not related to the classic Pringsheim’s example. Moreover, we observe that when only pairs of alternating series, both satisfying the assumptions of the alternating series test are considered, if one of them is absolutely convergent then the character of convergence of their Cauchy product is exactly the same as the character of convergence of the second factor. We complete the remarks with a new and surprisingly short proof of the Voss Theorem on Cauchy products.

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来源期刊

American Mathematical Monthly Mathematics-General Mathematics

CiteScore

0.80

自引率

20.00%

发文量

127

审稿时长

6-12 weeks

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