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Math and science::Analysis::Tao::08. Infinite sets

Bernhard Riemann's rearrangement theorem

Let \( \sum_{n=0}^{\infty} a_n \) be a series of reals which is conditionally convergent, but not absolutely convergent, and let L be any real number. Then there exists a bijection \( j : \mathbb{N} \rightarrow \mathbb{N} \) such that [...].

A supporting lemma can be defined to design the proof:

In a conditionally (but not absolutely) convergent series, both the sum of the positive terms and the sum of the negative terms are [...].