Series (mathematics)

In mathematics, a series is the cumulative sum of a given sequence of terms. Typically, these terms are real or complex numbers, but much more generality is possible.

For example, given the sequence of the natural numbers 1, 2, 3, ..., the series is 1, 1 + 2, 1 + 2 + 3, ...

According to the number of terms, the series may be finite or infinite. The former is relatively easy to deal with. In fact, the finite series is identified with the sum of all terms and — apart from the elementary algebra — there is no particular theory that applies. It turns out, however, that much care is required when manipulating infinite series. For example, some simple operations borrowed from elementary algebra — such as a change of order of the terms often lead to unexpected results. So it is sometimes tacitly understood, especially in analysis, that the term "series" refers to the infinite series. In what follows we adopt this convention and concentrate on the theory of the infinite case.

Motivations and examples

Given a series, an obvious question arises. Does it make sense to talk about the sum of all terms? Clearly, it is not always the case. Consider a simple example when the general term is constant and equal to 1, say. That is, the series reads as
1
2=1+1
3=1+1+1
...etc. One observes that the sum of all terms is not finite. In mathematical language, the series diverges to infinity (or it is divergent). There is not much to say about such object. If we want to build an interesting theory, that is to have some examples, operations and theorems, we need to deal with convergent series, that is series for which the sum of all terms is well-defined.

Actually, are there any? Maybe any series would diverge like this? Consider the following series of decreasing terms 1+1/2+1/4+1/8+1/16+... The sum is finite! Instead of a rigorous proof, we present a picture (see fig.1) which gives a geometric interpretation. Each term is represented by a rectangle of the corresponding area. At any moment (given any number of rectangular "chips on the table"), the next rectangle covers exactly one half of the remaining space. Thus, the chips will never cover more than the rectangle $2\times 1.$ In other words, the sum increases when more terms are added, but it does not go to the infinity, it never exceeds two. The series converges.

Formal definition

Given a sequence $a_{1},a_{2},\dots$ of elements that can be added, let

$S_{n}=a_{1}+a_{2}+\cdots +a_{n},\qquad n\in \mathbb {N} .$

Then, the series is defined as the sequence $\{S_{n}\}_{n=1}^{\infty }$ and denoted by $\sum _{n=1}^{\infty }a_{n}.$ For a single n, the sum $S_{n}$ is called the partial sum of the series.

If the sequence $(S_{n})$ has a finite limit, the series is said to be convergent. In this case we define the sum of the series as

$\sum _{n=1}^{\infty }a_{n}=\lim _{n\to \infty }S_{n}.$

Note that the sum (i.e. the numeric value of the above limit) and the series (i.e. the sequence $S_{n}$) are usually denoted by the same symbol. If the above limit does not exist - or is infinite - the series is said to be divergent.