Another chicken or egg: sequence or series

Question

This is a side question which is more motivated by teaching than research.

First, I am trying to convince myself that sequences appear before series (as numerical approximations to "interesting" quantities; on the other hand, decimal expansions -- especially infinite -- are more likely to be series).

Secondly, is it natural for sequences to be placed prior to series in a calculus course?

So, which one is more original, a sequence or a series?

After-dinner edit. We define a sequence to be... a function mapping the positive numbers to a set(?). We define a series to be... a formal infinite sum $\sum_{n=1}^\infty a_n$(?). Tell me what is your way to "define" these two guys, I do not believe they are very related.

There are no doubts that it is easier to define convergence of series via convergence of sequences, but it does not imply their "primogeniture". The notion of Cauchy sequence is an elegant way to build the apparatus of not only sequences but also of real numbers; as such it can serve as a definition of series: a series is a formal infinite sum $\sum_{n=1}^\infty a_n$, and it is called a convergent series if for any $\epsilon>0$ there exists an $N=N(\epsilon)$ such that for any $m>n>N$ the sum $|a_n+\dots+a_m|<\epsilon$. The real numbers then are nothing but representatives of equivalence classes of convergent series. (I have no desire here to expand all the details.) A sequence $b_n$ is convergent when the corresponding series $\sum_{n=1}^\infty a_n$ where $a_1=b_1$ and $a_n=b_n-b_{n-1}$ for $n\ge2$ converges. It would be honest to say that, besides the trivialities like "algebra of limits", the techniques for investigating convergence of series are quite independent from that of sequences. And it does not sound impossible to do series prior to sequences.

Historically, all these convergence/divergence issues were purely intuitive for both sequences and series, and they both were on the market for many centuries. I ask whether their exists an overwhelming historical support to the notion of sequence to lead.

Answer 1

This is not a precise answer, mainly some thoughts.

Historically both ideas seem very old (2000+ years), and sometimes it is hard to tell what point of view is predominant.

Say if one looks at the Method of Exhaustion or things around Zeno's paradoxes, what does arise, 'sequences' or 'series'? For some of the constructions, one seems more natural, for others the other. In any case, in some form of series already arose then.

One more point in favor of the fact that series were around early on: while in today's courses differentiation comes before integration, in an intuitive sense I think integration is rather the easier or more natural idea, and historically early forms of integration are very old (cf. Method of Exhaustion, which in my opinion counts as some sort of integration). And integration and series sort of go together.

For your second question, for a course today, I would however start with sequence.
If one wants to talk about series in a somewhat rigorous form just having the notion/word sequence at ones disposal already seems like a big plus.

Answer 2

I would argue that it is natural to introduce series first. Why are sequences interesting? The sequence $1$, $3/2$, $7/4$, etc. converges to $2$. Who cares?

I think the most natural answer to ``who cares'' is series. Write $e = 1 + 1 + 1/2 + 1/6 + \cdots$ on the blackboard, and I expect that students will know what is meant, and think it's cool. We write $1/3 = .3333\dots$ in precalculus courses without first discussing convergence, and this isn't really all that different.

Having introduced series, one can continue and write things like $1 - 1 + 1 - 1 + \cdots$ or whatever on the blackboard, and perhaps scare the students a little bit and explain that it is possible to write down formulas which are complete nonsense. (Or maybe only almost-complete nonsense, Ramanujan argued in cold blood that $1 + 2 + 3 + 4 + \cdots = -1/12$.) This motivates a more cautious approach to the subject, i.e. discussing convergence of sequences.

Answer 3

Most calculus students will see limits of sequences first, because definite integrals are limits of sequences of Riemann sums.

Answer 4

I have a rather radical idea. I start with Maclaurin Series! Let's see how it works. You first see graphically and "globally" that you get closer and closer to the function and when adding infinite terms you get the function. Then you have a point-wise look. For example, consider the Maclaurin series of Exp(x), you ask what happens at, say, x=1 (see the corresponding y-coordinates). Alongside "the convergent graphs", you have a numerical series. Playing with different functions, you get some interesting numerical series that without having any definition at hand it is not possible to decide whether they are convergent or not, ex. 1-1+1-1+1... It leads students to a definition of convergent numerical series and back again, convergent functional series! On the middle, we touch sequences.

It seems strange, but usually I have a big picture for each of my courses and this idea works well within the picture I have for Single Variable Calculus.

Answer 5

Why shouldn’t they come together? Here’s a possible program. The first object to appear, in an elementary course of analysis, if one starts by an axiomatic presentation of $\mathbb R$, is the supremum. It is simple to understand visually, and people can be easily convinced that it is a powerful existence principle, and serves to define a variety of useful derived objects, that will be studied and used throughout the course. One can list them as a program for the course: the sum of a family of non-negative numbers, the limit inferior and superior of sequences, the uniform norm (introducing the useful notation $\|f\|_{\infty, S}$, for a function $f:S\to \mathbb R$), the length or variation of a curve, the Riemann-Darboux integral.

The definition of limit of a sequence comes naturally generalising the property of a supremum (infimum) of an increasing (decreasing) sequence. The notion of sum of nonnegative numbers extends to a absolutely summable families, and immediately allows interesting computations like $\sum_{n\ge0} a^n$, $\sum_{n\ge0} na^n$, $\sum_{n\ge1} 1/n$,.., once one proves convenient rules of calculus, which are a toy-version of measure theory: generalized associativity, monotone convergence, dominated convergence, Fubini-Tonelli. By dominated convergence, one immediately has a double definition of $e^x=\lim_{n\to\infty}(1+x/n)^n=\sum_{k=0}^\infty x^k/k!$, expanding the binomial, a computation that can be resumed for $x\in\mathbb C$, without typographical changes.

Answer 6

Caveat: below I am only saying that it could be done, not that it should. I am in the nets/sequences first camp.

I think implicit in the sentiment behind this comment is

How can you define convergence of a series without referring to the convergence of its sequence of partial sums.

Because as we know, sequences are telescoping series are practically indistinguishable.

And I think in principle this could be done, and in some way can be a good motivation for the notion of completeness.

Essentially, one can start by motivating the idea that a series converges (settles down to a number) if its tail becomes "negligible", meaning that all the changes past a certain point is very small. This one can expand to meaning "past a certain point, the sum of any consecutive string of numbers in the series is no more than a fixed error". (Basically start by motivating, the definition for the series being Cauchy.)

Then one can start asking the question of whether there actually is a number that is being represented by this infinite sum (Cauchy completeness).

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BTW, thinking of a sequence as a telescoping series is occasionally useful in analysis (most textbook proofs of Banach fixed point, for example).