The convergence of a series \(\sum_{k=0}^\infty x_k\) to the limit \(L\) is a necessary but not a sufficient criterion for the convergence of any rearrangement of this series, formally
\[\begin{array}{l}\sum_{k=0}^\infty x_k=L\text{ convergent } \cancel\Longrightarrow (\text{ it follows not! })\\\sum_{k=0}^\infty x_{\sigma(k)}\text{ convergent for every permutation }\sigma:\mathbb N\to \mathbb N.\end{array}\]
Proofs: 1
Proofs: 1