Teoremas de reordenamiento de series

The sum of an infinite number of real numbers can depend on the arranging of these numbers. In this paper, we take you through several results about rearranging the terms of series; from series of real numbers to series in \(\\mathbb{R}^n\), and even results about series in Banach spaces. We do not include proofs of theorems but only their main ideas.

First, we study the real numbers series case, in which we see the Riemann rearrangement theorem** together with other results. We will continue with the **Lévy-Steinitz theorem, an analogous result of Riemann’s theorem for vector series in \(\\mathbb{R}^n\).

In particular, we will consider the Eisenstein series defined in the complex field. This series has the property that rearrangement in the order of summations results in a predictable change in the value of the series. This series is useful in the study of modular forms.

Finally, we show Pechersky’s theorem on rearrangement of series in Hilbert spaces, a result useful to prove the universality of the Riemann \(\\zeta\) function.

Recommended citation: Bello Hernández, M., Mahillo, A. (2020). Teoremas de reordenamiento de series. Zubía, 37-38, 129-148.
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