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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