11
Sequences and Series
Consider the following sum:
1
2
+
1
4
+
1
8
+
1
16
+
· · ·
+
1
2
i
+
· · ·
The dots at the end indicate that the sum goes on forever. Does this make sense? Can
we assign a numerical value to an infinite sum?
While at first it may seem difficult or
impossible, we have certainly done something similar when we talked about one quantity
getting “closer and closer” to a fixed quantity. Here we could ask whether, as we add more
and more terms, the sum gets closer and closer to some fixed value. That is, look at
1
2
=
1
2
3
4
=
1
2
+
1
4
7
8
=
1
2
+
1
4
+
1
8
15
16
=
1
2
+
1
4
+
1
8
+
1
16
and so on, and ask whether these values have a limit. It seems pretty clear that they do,
namely 1. In fact, as we will see, it’s not hard to show that
1
2
+
1
4
+
1
8
+
1
16
+
· · ·
+
1
2
i
=
2
i
−
1
2
i
= 1
−
1
2
i
233
234
Chapter 11 Sequences and Series
and then
lim
i
→∞
1
−
1
2
i
= 1
−
0 = 1
.
There is one place that you have long accepted this notion of infinite sum without really
thinking of it as a sum:
0
.
3333
¯
3 =
3
10
+
3
100
+
3
1000
+
3
10000
+
· · ·
=
1
3
,
for example, or
3
.
14159
. . .
= 3 +
1
10
+
4
100
+
1
1000
+
5
10000
+
9
100000
+
· · ·
=
π.
Our first task, then, to investigate infinite sums, called
series
, is to investigate limits of
sequences
of numbers. That is, we officially call
∞
summationdisplay
i
=1
1
2
i
=
1
2
+
1
4
+
1
8
+
1
16
+
· · ·
+
1
2
i
+
· · ·
a series, while
1
2
,
3
4
,
7
8
,
15
16
, . . .,
2
i
−
1
2
i
, . . .
is a sequence, and
∞
summationdisplay
i
=1
1
2
i
= lim
i
→∞
2
i
−
1
2
i
,
that is, the value of a series is the limit of a particular sequence.
While the idea of a sequence of numbers,
a
1
, a
2
, a
3
, . . .
is straightforward, it is useful to
think of a sequence as a function. We have up until now dealt with functions whose domains
are the real numbers, or a subset of the real numbers, like
f
(
x
) = sin
x
. A sequence is a
function with domain the natural numbers
N
=
{
1
,
2
,
3
, . . .
}
or the non-negative integers,
Z
≥
0
=
{
0
,
1
,
2
,
3
, . . .
}
. The range of the function is still allowed to be the real numbers; in
symbols, we say that a sequence is a function
f
:
N
→
R
. Sequences are written in a few
different ways, all equivalent; these all mean the same thing:
a
1
, a
2
, a
3
, . . .
{
a
n
}
∞
n
=1
{
f
(
n
)
}
∞
n
=1
As with functions on the real numbers, we will most often encounter sequences that
can be expressed by a formula. We have already seen the sequence
a
i
=
f
(
i
) = 1
−
1
/
2
i
,
11.1 Sequences
235
and others are easy to come by:
f
(
i
) =
i
i
+ 1
f
(
n
) =
1
2
n
f
(
n
) = sin(
nπ/
6)
f
(
i
) =
(
i
−
1)(
i
+ 2)
2
i
Frequently these formulas will make sense if thought of either as functions with domain
R
or
N
, though occasionally one will make sense only for integer values.
Faced with a sequence we are interested in the limit
lim
i
→∞
f
(
i
) = lim
i
→∞
a
i
.
We already understand
lim
x
→∞
f
(
x
)
when
x
is a real valued variable; now we simply want to restrict the “input” values to be
integers. No real difference is required in the definition of limit, except that we specify, per-
haps implicitly, that the variable is an integer. Compare this definition to definition 4.14.