In summary, "Series in Mathematics: From Zeno to Quantum Theory" explores the historical development and significance of mathematical series, tracing their origins from Zeno's paradoxes to their applications in modern quantum theory. The text examines how concepts of convergence, divergence, and infinite sums have evolved, illustrating their foundational role in calculus and mathematical analysis. It highlights key mathematicians and their contributions, emphasizing the impact of series on both theoretical and practical aspects of mathematics and physics.
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ToggleIntroductionZeno of Elea, Archimedes of Syracuse,
and John von Neumann of BudapestFunctionsConvergenceDomains of SeriesSources
Introduction
Series play a decisive role in many branches of mathematics. They accompanied mathematical developments from Zeno of Elea (##5##-th century BC) and Archimedes of Syracuse (##3##-th century BC), to the fundamental building blocks of calculus from the...

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Where is the quantum theory and where did you use reference [7]?
 
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martinbn said:
Where is the quantum theory and where did you use reference [7]?
Greg changed the title without asking me. My title was only "Mathematical Series". The word 'Mathematical' was already a concession. I would have called it just "Series".

It is possible to write an article with that actual title, and Dieudonné did write the middle part of such an article (17th to 19th century) but it took him several hundred pages - without Zeno, Archimedes, and Bohr.

I mentioned QM/Lie theory in the introduction and one can find a proof for Ad exp = exp ad in Varadarajan [7]. Maybe it should have been placed behind the formula, but I didn't want it to conflict with "(*)" which I needed for reference, so I chose the second-best location for [7].
 
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FAQ: Series in Mathematics: From Zeno to Quantum Theory

What is the historical significance of series in mathematics?

Series have played a crucial role in the development of mathematics, starting from ancient Greek philosophers like Zeno, who posed paradoxes that questioned the nature of motion and infinity. Over the centuries, mathematicians like Newton and Leibniz developed calculus, which formalized the concept of infinite series, paving the way for modern mathematics and physics, including the formulation of theories in quantum mechanics.

What are the different types of series in mathematics?

There are several types of series in mathematics, including arithmetic series, geometric series, and infinite series. An arithmetic series is the sum of the terms of an arithmetic sequence, while a geometric series involves terms that have a constant ratio. Infinite series, such as power series and Taylor series, extend these concepts to infinitely many terms and are fundamental in calculus and analysis.

How do series converge or diverge?

The convergence or divergence of a series is determined by the behavior of its terms as they approach infinity. A series converges if the sum of its terms approaches a finite limit, while it diverges if the sum grows without bound or does not settle at a specific value. Various tests, such as the ratio test, root test, and comparison test, are used to analyze convergence properties.

What is the relationship between series and calculus?

Series are intimately connected to calculus, particularly in the study of functions and their approximations. The concept of a Taylor series allows for the representation of functions as infinite sums of terms calculated from the values of their derivatives at a single point. This representation is essential for solving differential equations and for numerical methods in calculus.

How are series applied in quantum theory?

In quantum theory, series are used to solve complex problems involving wave functions and probabilities. Techniques such as perturbation theory rely on series expansions to approximate the behavior of quantum systems in the presence of small disturbances. Additionally, Fourier series are employed to analyze wave functions, leading to insights about the nature of particles and their interactions.

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