MATH 118, LECTURES 16, 17 & 18:
SEQUENCES
1
Sequences
Many processes in mathematics and the applied sciences result in
sequences
of numbers.
Consider many of the processes of approximation we have carried out in
this and the previous semester’s Calculus course. For example, consider the
Newton’s method
formula
x
n
+1
=
x
n

f
(
x
n
)
f
prime
(
x
n
)
used to estimate the roots of a function (i.e.
x
*
∈
R
such that
f
(
x
*
) = 0).
What this formula generated was a sequence of estimates for the roots which
could have been written as
{
x
1
, x
2
, x
3
, x
4
, . . .
}
=
{
x
n
}
∞
n
=1
(1)
where the dots represent the fact that the sequence does not terminate at
some finite index
n
—it is an
infinite
sequence.
Similarly, when estimating the area under the curve for functions we
could not integrate, we applied various numerical integration techniques to
obtain estimates for the area. This could be seen as generating a sequence
of numbers; for instance, using the Rectangular Rule we have
I
n
=
h
n
summationdisplay
i
=1
f
(
x
i
) =
(
b

a
)
n
n
summationdisplay
i
=1
f
parenleftbigg
a
+
(
b

a
)
n
i
parenrightbigg
which generates the infinite sequence
{
I
1
, I
2
, I
3
, I
4
, . . .
}
=
{
I
n
}
∞
n
=1
.
(2)
In both of these cases, we were hopeful that the estimates
x
n
and
I
n
became closer and closer approximates to the root of
f
(
x
) and the actual
area under the curve, respectively, as
n
→ ∞
. In terms of the sequences (1)
and (2), we were interested in whether or not they
converge
. If we take
x
*
to be the actual value of the root (
f
(
x
*
) = 0) and
I
*
to be the actual value
1
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of the area under the curve, we represent that the sequences converge to the
limits
x
*
and
I
*
respectively by
x
n
→
x
*
,
I
n
→
I
*
as
n
→ ∞
or
x
n
n
→∞
→
x
*
,
I
n
n
→∞
→
I
*
or
lim
n
→∞
x
n
=
x
*
,
lim
n
→∞
I
n
=
I
*
.
Whether infinite sequences of numbers converge or not will be an issue of
central important to us. (We will discuss the mathematical formulation of
convergence a little later on.)
There is another important distinction to be drawn from the examples
above. The approximate area formula
I
n
depended only on the value of
n
,
so that if we wanted to find, say,
I
100
, we could simply plug
n
= 100 into
the formula. We did not need any information whatsoever about the values
of
I
n
for
n
= 1
, . . . ,
99—we could simply jump to consideration of the 100
th
term is the sequence.
Sequences defined this way are said to be defined
explicitly
.
Now consider the formula for the estimates of a root given by
x
n
. In order
to evaluate the 100
th
number in the sequence, we first need to calculate the
99
th
term in the sequence, which depends on the 98
th
term, and so on.
In fact, we cannot evaluate the estimate at any value of
n
without having
calculated all preceding values in the sequence! Sequences defined in such a
manner are said to be defined
recursively
.
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 Spring '09
 Soares

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