Series in Mathematics: From Zeno to Quantum Theory

[fresh_42]

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 $\displaystyle{\sum_{n=1}^\infty a_n=c}$...

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[martinbn]

Where is the quantum theory and where did you use reference [7]?

 

[fresh_42]

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].