Series (mathematics)

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of

calculus

and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as

physics

, computer science,

statistics

and

finance

.

For a long time, the idea that such a potentially infinite

summation

could produce a finite result was considered

paradox

ical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on.

Zeno

concluded that Achilles could ''never'' reach the tortoise, and thus that movement does not exist. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise.

In modern terminology, any (ordered) infinite sequence $(a_1,a_2,a_3,\ldots)$ of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the one after the other. To emphasize that there are an infinite number of terms, a series may be called an infinite series. Such a series is represented (or denoted) by an expression like
$a_1+a_2+a_3+\cdots,$
or, using the summation sign,
$\sum_^\infty a_i.$

The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as tends to

infinity

(if the limit exists) of the finite sums of the first terms of the series, which are called the th partial sums of the series. That is,
$\sum_^\infty a_i = \lim_ \sum_^n a_i.$
When this limit exists, one says that the series is convergent or summable, or that the sequence $(a_1,a_2,a_3,\ldots)$ is summable. In this case, the limit is called the sum of the series. Otherwise, the series is said to be divergent.

The notation $\sum_^\infty a_i$ denotes both the series—that is the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the result of the process. This is a generalization of the similar convention of denoting by $a+b$ both the

addition

—the process of adding—and its result—the ''sum'' of and .

Generally, the terms of a series come from a ring, often the field $\mathbb R$ of the real numbers or the field $\mathbb C$ of the

complex number

s. In this case, the set of all series is itself a ring (and even an associative algebra), in which the addition consists of adding the series term by term, and the multiplication is the Cauchy product.

Basic properties

An infinite series or simply a series is an infinite sum, represented by an infinite expression of the form
$a_0 + a_1 + a_2 + \cdots,$
where $(a_n)$ is any ordered

sequence

of terms, such as

number

s, functions, or anything else that can be

added

(an abelian group). This is an expression that is obtained from the list of terms $a_0,a_1,\dots$ by laying them side by side, and conjoining them with the symbol "+". A series may also be represented by using summation notation, such as
$\sum_^ a_n .$

If an abelian group of terms has a concept of limit (e.g., if it is a metric space), then some series, the convergent series, can be interpreted as having a value in , called the ''sum of the series''. This includes the common cases from

calculus

, in which the group is the field of real numbers or the field of

complex number

s. Given a series $s=\sum_^\infty a_n$, its th partial sum is
$s_k = \sum_^a_n = a_0 + a_1 + \cdots + a_k.$
By definition, the series $\sum_^ a_n$ ''converges'' to the limit (or simply ''sums'' to ), if the sequence of its partial sums has a limit . In this case, one usually writes
$L = \sum_^a_n.$
A series is said to be ''convergent'' if it converges to some limit, or ''divergent'' when it does not. The value of this limit, if it exists, is then the value of the series.

Convergent series

A series is said to converge or to ''be convergent'' when the sequence of partial sums has a finite limit. If the limit of is infinite or does not exist, the series is said to diverge. When the limit of partial sums exists, it is called the value (or sum) of the series
$\sum_^\infty a_n = \lim_ s_k = \lim_ \sum_^k a_n.$

An easy way that an infinite series can converge is if all the are zero for sufficiently large. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.

Working out the properties of the series that converge, even if infinitely many terms are nonzero, is the essence of the study of series. Consider the example
$1 + \frac+ \frac+ \frac+\cdots+ \frac+\cdots.$
It is possible to "visualize" its convergence on the real number line: we can imagine a

line

of length 2, with successive

segment

s marked off of lengths 1, 1/2, 1/4, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: When we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is ''equal'' to 2 (although it is), but it does prove that it is ''at most'' 2. In other words, the series has an upper bound. Given that the series converges, proving that it is equal to 2 requires only elementary algebra. If the series is denoted , it can be seen that
$S/2 = \frac = \frac+ \frac+ \frac+ \frac +\cdots.$
Therefore,
$S-S/2 = 1 \Rightarrow S = 2.$

The idiom can be extended to other, equivalent notions of series. For instance, a recurring decimal, as in
$x = 0.111\dots ,$
encodes the series
$\sum_^\infty \frac.$

Since these series always converge to

real numbers

(because of what is called the completeness property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, the decimal expansion 0.111... can be identified with 1/9. This leads to an argument that , which only relies on the fact that the limit laws for series preserve the arithmetic operations; for more detail on this argument, see 0.999....

Examples of numerical series

* A ''geometric series'' is one where each successive term is produced by multiplying the previous term by a constant number (called the common ratio in this context). For example: $1 + + + + + \cdots=\sum_^\infty = 2.$ In general, the geometric series $\sum_^\infty z^n$ converges if and only if $, z, < 1$, in which case it converges to harmonic series