Rearrangement of conditionally convergent series on a small set

Rearrangement of a Conditionally Convergent Series

One of the surprising results in an elementary calculus course is that a rearrangement of a conditionaly convergent series may change its sum, even its very convergence. Observing typical textbook examples of this phenomenon, it turns out that during the rearrangement some of the terms are moved arbitrarily large distances from their original locations. Is this necessary? The answer is positive...

Changes of signs in conditionally convergent series on a small set

We consider ideals I of subsets of the set of natural numbers N such that for every conditionally convergent series of real numbers ∑ n∈N an and s ∈ R, then there is a sequence of signs δ = (δn)n∈N such that ∑ n∈N δnan = s and N(δ) := {n ∈ N : δn = −1} ∈ I. We give some properties of such ideals and characterize them in terms of extendability to a summable ideal.

On rearrangements of conditionally convergent series

A Distributional Approach to Conditionally Convergent Series

Whether the car’s gas tank is filled up on Monday and the paycheck is deposited on Tuesday, or vice versa, the contribution of those two transactions to the checkbook’s final balance is the same. By the commutative property, order does not matter for the algebraic addition of a finite number of terms. However, for a super banker who conducts an infinite number of transactions, order may matter....

Large Partial Random Permutations of Conditionally Convergent Series

This paper studies the behavior of sums of series under random permutations. The first result considers the case of uniformly distributed permutations on the set N of all integers, which requires finitely additive probability. The next result considers the limit of the first portion p, 0 ≤ p ≤ 1, of a uniform distribution on the first n integers, as n gets large. The focal point is when 0 < p <...