Rearrangement of a Conditionally Convergent Series

Rearranging Series Constructively

Riemann’s theorems on the rearrangement of absolutely convergent and conditionally convergent series of real numbers are analysed within Bishop-style constructive mathematics. The constructive proof that every rearrangement of an absolutely convergent series has the same sum is relatively straightforward; but the proof that a conditionally convergent series can be rearranged to converge to what...

On rearrangements of conditionally convergent series

Computing sums of conditionally convergent and divergent series using the concept of grossone

Let a1, a2, . . . be a numerical sequence. In this talk we consider the classical problem of computing the sum ∑∞ n=1 an when the series is either conditionally convergent or divergent. We demonstrate that the concept of grossone, proposed by Ya. Sergeyev in [1], can be useful in both computing this sum and studying properties of summation methods. First we prove that within the grossone univer...

ar X iv : h ep - t h / 04 10 07 1 v 2 3 1 D ec 2 00 4 DIVERGENCE OF THE 1 N f - SERIES EXPASION IN QED

The perturbative expansion series in coupling constant in QED is divergent and is expected to be, at the best, conditionally convergent. The 1/Nf series expansion, where Nf is the number of flavours, defines a rearrangement of this perturbative series, and therefore, its convergence would serve as a proof that the perturbative series is, in fact, conditionally convergent.Unfortunately, the 1/Nf...

The Descriptive Complexity of Series Rearrangements

We consider the descriptive complexity of some subsets of the infinite permutation group S∞ which arise naturally from the classical series rearrangement theorems of Riemann, Levy, and Steinitz. In particular, given some fixed conditionally convergent series of vectors in Euclidean space Rd, we study the set of permutations which make the series diverge, as well as the set of permutations which...