# Another Chicken or Egg: Sequence or Series

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.

• We define series using sequences. How would you define sequences using series? Aug 26 '12 at 10:19
• "Tell me what is your way to "define" these two guys, I do not believe they are very related." In my first calculus course (still not sure whether I should refer to it as a calculus, or a real analysis course, though) we used the following definition of series: Let there be a sequence $(a_n)$ in a normed space $X$ and let $s_k=\sum_{n=1}^k a_n$ for $n\in\mathbb{N}$. Series in $X$ is an ordered pair $(a_n,s_k)$ ($a_n,s_k\in X$) which is consisted of two sequences $(a_n)$ and $(s_k)$, former being the terms, and latter being partial sums of series. Aug 26 '12 at 12:24
• should be CW Aug 26 '12 at 18:22
• @HarunŠiljak, re, if your sequences $(s_n)_n$ are valued in groups, then they are precisely the series $s_0 + (s_1 - s_0) + (s_2 - s_1) + \dotsb$, no? Jun 4 at 15:04

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, the one seems more natural for others the other. In any case, in some form series already arose then.

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

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.

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.

• really? series are just situations where elements of a sequence get added. we could be doing other stuff like multiplication, exponentiation, transformation, etc. etc., with elements of a sequence---so I don't really agree with it is "natural" to introduce series first...what if the elements of our sequence do not come from a space where addition is defined? we can still have sequences.... Aug 26 '12 at 14:51
• This is a calculus class. Addition is defined. Aug 26 '12 at 15:06
• (I know, I was just being snide because of the "natural" in there) Aug 26 '12 at 15:10
• I definitely like your point, Frank. Your example with $e$ could be acomplished with the limit of $(1+1/n)^n$ as $n\to\infty$, which is extremely useful in showing many other limits but at the same time impractical for actual computation of $e$. The series, on other hand, can be successfully used not only to compute the number but also to demonstrate its irrationality to a first year undergrad. Aug 27 '12 at 9:36
• I am pretty sure that we should give the credit of that "almost-complete nonsense" you have mentioned to Euler :) May 5 '13 at 19:34

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.

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