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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 ##17##-th century on, up to modern Lie theory which is crucial for our understanding of quantum theory. Series are probably the second most important objects in mathematics after functions. And the latter have often been expressed by series, especially in analysis. The term analytical function or holomorphic function represents such an identification.
A series itself is just an expression
$$
\sum_{n=1}^\infty a_n =a_1+a_2+\ldots+a_n+\ldots
$$
but this simple formula is full of possibilities. It foremost contains some more or less obvious questions:
Do we always have to start counting at one?
What does infinity mean?
Where are the ##a_n## from?
Can we meaningfully assign a value ##\displaystyle{\sum_{n=1}^\infty a_n=c}##...
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".martinbn said:Where is the quantum theory and where did you use reference [7]?
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.
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.
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.
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.
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.