# What is the Sum of an Infinite Series?

## Finite Series

We first learn how to add two numbers at a time. Once we have learned
the associative law for addition
(x + y) + z = x + (y + z)

we understand that it makes sense to form the sum of any finite list
of numbers. The sum

u_1 + u_2 + . . .
+ u_N

of a finite list of numbers is sometimes called a finite sum or the
sum of a finite series.

## Series v. Sequence

A *sequence* is a simply an ordered collection of numbers
a_1, a_2, a_3,
. . .

that is indexed by positive integers. If the terms

a_j

are *real*, then the
sequence is called a *sequence of real numbers*.

Frequently in this part of the course we shall want to discuss
*sequences of complex numbers*, i.e., sequences in which each
term a_j is a complex number

a_j = b_j + i c_j

where b_j and c_j are real and i = sqrt(-1).

A *series*

u_1 + u_2 +u_3
+ . . .

is a *formal expression* that is intended to represent the
*sum* of its sequence of *terms*

u_1, u_2, u_3,
. . . .

The terms u_j may be real or complex numbers.

## Sums

But **what is the ***sum*?
If the series is a finite series, then the meaning of its sum is
known from elementary mathematics.

If the series is infinite, i.e., involves infinitely many terms,
then the meaning of *sum* is **not** known from elementary
mathematics and requires careful, precise specification.

That is what the study of infinite series is about.

There is more than one approach.

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