1
Study Guide for Prelim 1
Math 192  Spring 1997
Written by Don Allers; Revised by Sean Carver
1
Improper Integrals
Idea:
A proper integral is an integral
R
b
a
f
(
x
)
dx,
where
a
and
b
are both finite and
f
(
x
) is continuous and
finite on [
a, b
]. For proper integrals, the
fundamental theorem of calculus
tells us that if
F
is an antiderivative
of
f
(i.e. if F’(x) = f(x)) then
R
b
a
f
(
x
)
dx
=
F
(
b
)

F
(
a
)
.
We will define an improper integral as a limit of
proper integrals.
Comment
The integral
R
b
a
f
(
x
)
dx
is
improper
if: (i)
a
or
b
is infinite, or (ii)
f
(
x
) is not finite on all of
[
a, b
]. The fundamental theorem of calculus does not apply directly to this class of integrals. Indeed, the
definition of a definite integral given in Chapter 4 explicitly assumes that the integral is proper. Without
the definitions below, an improper integral would be an expression with no meaning.
Solution
Method:
1. If
a
is finite, and
f
(
x
) is continuous on [
a,
∞
] we define
R
∞
a
f
(
x
)
dx
= lim
c
→∞
R
c
a
f
(
x
)
dx.
2. If
b
is finite, and
f
(
x
) is continuous on [
∞
, b
] we define
R
b
∞
f
(
x
)
dx
= lim
c
→∞
R
b
c
f
(
x
)
dx.
3. If
a, b
are finite, and
f
(
x
) is continuous on (
a, b
] we define
R
b
a
f
(
x
)
dx
= lim
c
→
a
+
R
b
c
f
(
x
)
dx.
4. If
a, b
are finite, and
f
(
x
) is continuous on [
a, b
) we b define
R
b
a
f
(
x
)
dx
= lim
c
→
b

R
c
a
f
(
x
)
dx.
5. If the integral is improper at multiple points on the domain [
a, b
], we divide this domain into
subintervals [
a, c
1
]
,
[
c
1
, c
2
]
, . . . ,
[
c
n

1
, c
n
]
,
[
c
n
, b
] so that each subinterval contains only one problem
point at one of its endpoints, and no problem points in its interior.
Then define
R
b
a
f
(
x
)
dx
=
R
c
1
a
f
(
x
)
dx
+
R
c
2
c
1
f
(
x
)
dx
+
. . .
R
c
n
c
n

1
f
(
x
)
dx
+
R
b
c
n
f
(
x
)
dx,
applying the above rules to each piece.
Eg:
(a)
Z
1
0
x
ln
x dx,
(b)
Z
∞
0
dx
x
2
+ 1
,
(c)
Z
∞
∞
2
x dx
(
x
2
+ 1)
2
,
(d)
Z
1

1
dx
x
2
/
3
.
2
Sequences
Idea:
Given a sequence of numbers
{
a
n
}
=
a
1
, a
2
, a
3
, . . .
, the question is whether or not these numbers are
converging
to some
limit
a
. We say that
{
a
n
}
converges to
a
, written
a
n
→
a
, as
n
→ ∞
, if the distance

a
n

a

between the sequence points and
a
goes to zero as
n
increases.
Extend the Sequence to a Real Function
Idea:
Compute the limit of a sequence as the limit of a function
f
(
x
) as
x
→ ∞
.
When?
Use this method if the expression defining the sequence
a
n
can be considered as a real
function in the variable
n
whose limit as
n
→ ∞
can be computed easily.
Method:
To compute the limit of the sequence, take take the limit of its defining expression as you
would take the limit of a real function. Remember L’Hopital’s Rule. (The theorem behind this method
is the following: If
f
(
n
) =
a
n
and lim
x
→∞
f
(
x
) =
L
then lim
n
→∞
a
n
=
L.