G
eometric
S
eries
Before we define what is meant by a series, we need to introduce a related topic, that
of sequences. Formally, a
sequence
is a function that computes an ordered list. Suppose
that on day 1, you have 1 dollar, and every day you double your money. Then the
function
f
(
n
) = 2
n
generates the sequence
1, 2, 4, 8, 16, 32, …,
when
n
= 1, 2, 3, 4, 5, 6, … This list represents the amount of dollars you have after
n
days. Note: The use of “…” is read as “and so on”.
The individual entries in a sequence are called the
terms
of the sequence. In our
discussion, we are going to assume that the terms in a particular sequence are real
numbers.
Sequences can be grouped into two large classes based upon the number of terms they
include. An
infinite sequence
is a function that has the set of natural numbers as its
domain. As the name implies, it contains an infinite number of terms. In the opening
example, the use of the “…” without some number on the end implies that the sequence
continues indefinitely, following the prescribed pattern.
Of course, there is an inherent problem with assuming that money can be doubled
forever. Instead, it makes sense to talk about doubling money for a certain number of
days. Say, for
n
= 1, 2, 3, 4, 5, 6, and 7. In that case, the sequence generated would be
called a
finite sequence
. Its domain is equal to a finite set of natural numbers. (In this
case,
D
= {1, 2, …, 7}.)
A common notation for sequences is let
a
n
=
f
(
n
). With this notation, we say that
a
n
is
the
n
th
term in the sequence.
Example 1:
Write out the first five terms
a
1
,
a
2
,
a
3
,
a
4
and
a
5
of the following sequences.
(a)
(1
)
n
n
a
(b)
2
sin
n
an
(c)
25
n
(d)
1
2(3)
n
n
a
1