Calculus: Analytic Geometry and Calculus, with Vectors

5.6 Sequences, series, and decimals 335

represent felons in courts of law, but nevertheless our precious corpus of
scientific information is not appreciably augmented when a solemn tutor
makes the unexplained statement that "decimals represent numbers."
To attack this and other matters, we must learn about some things that
have many applications. A sequence s1i s2, s3, of numbers is an
ordered collection of numbers in which there is a first, a second, a third,
etcetera. The individual numbers are called elements of the sequence;
they are not called terms because terms are things that are added, and
they are not called factors because factors are things that are multiplied.
When S1, S2, s3,. .is a given sequence, it may be true (or it may be
false) that there is a number L such that sn is near L whenever n is large.
This statement is meaningful. It means that when s1, S2, S3, is a
given sequence, it may be true (or it may be false) that there is a number
L such that to each e > 0 there corresponds an integer N such that
is, - LI < e whenever n > N. In case L exists, we write

lim sn = L,

n-+ m

as in Section 3.3, and we say that the sequence converges to L. In case the
limit does not exist, we say that the sequence is nonconvergent or divergent.
As we shall see, the elementary theories of sequences and series are
closely related. However, a series is very different from a sequence. A
series (or simple infinite series) is an array of numbers and plus signs of the
form

(5.61) u1 + u2 + U3 +...

Because the notion of addition is involved, the numbers ul, u2, ua, '

are called terms of the series. The terms are not necessarily nonnegative,

and it is standard practice to write the series

1+(-i)+ -+(-1)+- +(-)+ ...

in the form

1-- +f+*-*+....

Our series ul + u2 + u3 + contains so many terms that not even a

high-speed electronic computer could "add them all up" during its life-

time. In order to determine a number that can reasonably be called the

value of the series, we need a procedure involving more than brute-force

addition. While other procedures exist and are useful, the following is the

most elementary and best known useful procedure. Let the sequence

s1, $2, S3, of partial sums be defined by the formulas s1 = ul,
J2 = u1 + u2, S3 = U1 + u2 + u3, etcetera, so that

n

(5.62) Sn = u1 + U2 + + U. = I uk (n=1,2,3,

k-1