论文标题
发散序列的收敛性子系
Convergent subseries of divergent series
论文作者
论文摘要
令$ \ mathscr {x} $为正真实序列$ x =（x_n）$，这样系列$ \ sum_n x_n $是发散的。对于\ Mathscr {X} $中的每个$ X \，令$ \ Mathcal {i} _x $是所有$ a \ subseteq \ subseteq \ mathbf {n} $的集合，以使subseries $ \ sum_ {n \ in a}} x_n $ in Conventent。此外，让$ \ Mathscr {a} $是\ Mathscr {X} $的序列集$ x \，这样，$ \ lim_n x_n = 0 $ and $ \ \ \ \ \ m neq \ neq \ neq \ neq \ neq \ nec \ nec \ nec \ nec \ nec \ nec {i} $ \ liminf_n y_ {n+1}/y_n> 0 $。我们表明$ \ mathscr {a} $是合并的，其中包含许多序列$ x $，它们会生成成对的非构态理想$ \ MATHCAL {i} _x $。尤其是M. Filipczak和G. Horbaczewska最近提出的一个空旷的问题。
Let $\mathscr{X}$ be the set of positive real sequences $x=(x_n)$ such that the series $\sum_n x_n$ is divergent. For each $x \in \mathscr{X}$, let $\mathcal{I}_x$ be the collection of all $A\subseteq \mathbf{N}$ such that the subseries $\sum_{n \in A}x_n$ is convergent. Moreover, let $\mathscr{A}$ be the set of sequences $x \in \mathscr{X}$ such that $\lim_n x_n=0$ and $\mathcal{I}_x\neq \mathcal{I}_y$ for all sequences $y=(y_n) \in \mathscr{X}$ with $\liminf_n y_{n+1}/y_n>0$. We show that $\mathscr{A}$ is comeager and that contains uncountably many sequences $x$ which generate pairwise nonisomorphic ideals $\mathcal{I}_x$. This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska.
