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Reconciling Reimann's Series Theorem
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[QUOTE="gopher_p, post: 4880212, member: 414293"] The following are provable theorems: 1. If ##\sum a_n## is convergent but not absolutely convergent, then for all ##r\in\mathbb{R}##, there is a permutation ##\sigma## of ##\mathbb{N}## such that ##\sum a_{\sigma(n)}=r##. 2. If ##\sum a_n## is convergent and ##\sigma## is a permutation of ##\mathbb{N}## such that there is ##M\in\mathbb{N}## with ##|\sigma(n)-n|\leq M## for all ##n##, then ##\sum a_{\sigma(n)}## is convergent and ##\sum a_{\sigma(n)}=\sum a_n##. Your result relies on permutations that are "bounded" like those in (2). [/QUOTE]
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Reconciling Reimann's Series Theorem
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