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 Relative Articles Banishing divergence Part 1: Infinite numbers as the limit of sequences of real numbers Sequences and Limits Oscillatory Nonautonomous Lucas Sequences Sequences of knots and their limits Limits of zeros of polynomial sequences Limits of Sequences of Markov Chains Limits of dense graph sequences Limits of randomly grown graph sequences Limits of permutation sequences Sequences of Inequalities Among New Divergence Measures More...
Mathematics  2011

# Banishing divergence Part 2: Limits of oscillatory sequences, and applications

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Abstract:

Sequences diverge either because they head off to infinity or because they oscillate. Part 1 \cite{Part1} of this paper laid the pure mathematics groundwork by defining Archimedean classes of infinite numbers as limits of smooth sequences. Part 2 follows that with applied mathematics, showing that general sequences can usually be converted into smooth sequences, and thus have a well-defined limit. Each general sequence is split into the sum of smooth, periodic (including Lebesgue integrable), chaotic and random components. The mean of each of these components divided by a smooth sequence, or the mean of the mean, will usually be a smooth sequence, and so the oscillatory sequence will have at least a leading term limit. Examples of limits of oscillatory sequences with well-defined limits are given. Methodologies are included for a way to calculate limits on the reals and on complex numbers, a way to evaluate improper integrals by limit of a Riemann sum, and a way to square the Dirac delta function.

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