单调级数子级数收敛性的稀释判别法
A Dilution Test for the Convergence of Subseries of a Monotone Series
摘要：
柯西凝聚检验允许通过查看仅涉及由2的幂索引的原始级数的项的加权子级数来确定单调级数的收敛性。人们自然会问，反过来是否也是正确的：是否有可能通过查看原始级数的适当加权版本来确定单调级数的任意子级数的收敛性？在本说明中，我们表明，答案是肯定的，并介绍了一个新的收敛性测试，特别是为此目的而设计的。
英文摘要：
Cauchy's condensation test allows to determine the convergence of a monotone series by looking at a weighted subseries that only involves terms of the original series indexed by the powers of two. It is natural to ask whether the converse is also true: Is it possible to determine the convergence of an arbitrary subseries of a monotone series by looking at a suitably weighted version of the original series? In this note we show that the answer is affirmative and introduce a new convergence test particularly designed for this purpose.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
