(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.
a_1, a_2, a_3, . . .
that is indexed by positive integers. If the terms
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).
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.
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.