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

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.

• A "finite expression" to "series" is like $\pi$ to $3.1416$. Aug 17, 2015 at 11:36
• @ChristianBlatter I don't get it. What's the conclusion? Aug 17, 2015 at 12:18

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)
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.