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Comment by m4x | Hacker News Reader

Strictly speaking they aren't a "part of calculus" - they aren't directly related to continuous rates of change - but that's a pretty arbitrary distinction and both are almost always taught as part of a calculus course.

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Stewart defines calculus as the study of limits, which include infinite series.

In my opinion a more all-encompassing (if also more obtuse) definition would be: Calculus is the study of completeness.

I rather like this phrasing because basically everything you'll learn in calculus is a consequence of the completeness property of the reals.

How would you define and reason about integration without infinite series? Genuinely curious if someone has tried to do that.

I'm pretty sure that's not possible, since the integral is defined as a limit of Riemann sum.

There are actually a number of different ways to define integrals - see https://en.m.wikipedia.org/wiki/Integral#Formal_definitions

It's usually not really defined as the limit of a series in analysis texts, but as the simultaneous greatest lower bound and least upper bound of upper and lower sums (inscribed and circumscribed rects area), respectively. THat def is a lot easier to deal with in proofs and just as intuitive to many.

For many people, integration is defined as "the area under the curve" and computed using sets of known rules. Infinite series are a tool we use to reason about calculus concepts more formally. It's a very arbitrary distinction; I was only making it because if you aren't studying calculus in depth you can avoid talking about Taylor polynomials and infinite series, but you can't really avoid talking about derivatives and integrals. If you want to define integrals formally, then you will definitely see infinite series.

Although there are a few ways to define the reals, one way is to define it as the limit of an infinite rational series.

A good def for the R. integral is: the one number which is both a lower bound of the set of all upper sums (area with superscribed rectangles), and an upper bound of all lower sums (inscribed rects).

That has a lot of nice properties which are useful in proofs, whereas the limit of sums definition suffers from there being too many ways for "delta x goes to 0" which need to be shown to be equivalent at some point to be useful (nasty!).

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