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It seems that historically, there were two trends on the idea of integration:

  • Newton's work which depended on infinite series.
  • Leibniz work which depended on the dream of integration of elementary functions in a finite combination of basic functions which was later proved to be impossible by Liouville. I'll call it "calculus on finite expressions".

I took a course of calculus and it seems that integration via Leibniz way becomes increasingly complicated as the course progress. I didn't have a course on series yet but it seems that most of the functions can be represented as infinite power series and the integration on power series (integrating term by term) is often an easier task.

Until now, I guess I had more evidence that calculus on series is way more simpler and powerful than the calculus on finite expressions. My doubts are:

  • Is this correct?

  • Why don't we start studying calculus via series instead of the calculus on finite expressions?

Most of the books authors seem to think different about this matter because of their choice of order in the subjects. But Kuratowski's: Introduction to Calculus starts already with sequences and series, so I guess that perhaps that claim could be true. Although there is also another hypothesis: He could be lecturing in an educational system in which the calculus of finite expressions was taught early in high-school.

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  • $\begingroup$ A "finite expression" to "series" is like $\pi$ to $3.1416$. $\endgroup$ – Christian Blatter Aug 17 '15 at 11:36
  • $\begingroup$ @ChristianBlatter I don't get it. What's the conclusion? $\endgroup$ – Billy Rubina Aug 17 '15 at 12:18
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Even if one starts studying via series (And some courses do), one usually doesn't go to term-by-term integration and differentiation for a few reasons:

  1. It doesn't always work, and you have to deal with convergence issues, and uniform convergence, and this just makes it more difficult at times, especially for a student who is just starting.
  2. It is not always easy to identify a power series. You may get a power series which you don't know how to identify, and don't know how it behaves, but if you were to perform the calculation without using a series, you would get an expression which you have some intuition about. (e.g. $x\sin(x)$, integration would give $\sin(x)-x\cos(x)$, which is easier to figure out how it behaves without the series)
  3. You would still have to learn integration of rational functions, as using power series for them requires (At times) multiplication of function series, which is not nice, and gives no knowledge of the result.
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Term-by-term integration and differentiation of series is very useful when it works, but nailing down precisely when it does and doesn't work needs more mathematical sophistication than starting to learn calculus. From the point of view of applications (i.e. science and engineering as compared with mathematics), learning how to use "dangerous" mathematical tools without understanding the dangers is probably not a good idea.

On the other hand, you could make a case for starting with so-called "non-standard analysis" which makes the concept of "infinitesimals" rigorous, and thus puts the hand-wavy arguments of Newton, Leibniz, Euler, etc on a rigorous foundation. This avoids the use of limits and $\epsilon$-$\delta$ definitions.

See for example http://www.math.wisc.edu/~keisler/calc.html, or https://en.wikipedia.org/wiki/Non-standard_calculus (and the links on that page) for an overview.

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