Math 1220-002
Camacho
Fall 2012
Midterm #2 Review
Partial Fraction Decomposition

Distinct Linear Factors:

Repeated Linear Factors:

Quadratic Factors:
Integration Techniques



Integration By Parts (and Tabular Integration)
U-Substitution
Partial Fraction Decomposition
Indeterminate Forms

L’Hopital’s Rule:
provided that either….
o
OR
o

Types of Indeterminate Forms:
o
o
o
or
or
or
or
Apply L’Hopital’s Rule directly.
Convert to fraction and Apply L’Hopital’s Rule
Take natural log, simplify, and apply L’Hopital’s Rule
Math 1220-002
Camacho
Fall 2012
Midterm #2 Review
Improper Integrals



If
has an asymptote at
, where is in the interval
then
Infinite Sequences

represents the infinite sequence of numbers
, where

The limit of the sequence
is defined as
.
o The sequence converges if exists and is finite
o The sequence diverges if does not exist or is infinite.

If
and the sequences
and
squeeze theorem,
must converge to .

If sequences are monotonic and bounded, then…
o If
is monotonically increasing and bounded above, then it must converge.
o If
is monotonically decreasing and bounded below, then it must converge.
both converge to , then by the
Infinite Series

An infinite series is the sum of all terms in an infinite sequence

The th partial sum in a sequence is the sum of the first
terms.
Math 1220-002
Camacho

Fall 2012
Midterm #2 Review
A geometric series has the form
o If
, then
o If
, then
.
diverges
o

The term test says that if

The integral test says that

A p-series,

The comparison test says that if
o If
converges, then
o If
diverges, then

The limit comparison test says that if
o
o

, then the series diverges.
converges if and only if
, converges when
and
and
and diverges when
where
, then either
both converge OR
both diverge
, then
then
converges
then
diverges
then the test is inconclusive.

The alternating series of the sequence

An alternating series converges if

A series is absolutely convergent if

A series is conditionally convergent if

The absolute ratio test says that if
o If
o If
o If
.
, then
converges
diverges
The ratio test says that if
o If
o If
o If
converges.
is defined as
.
converges
converges but
the series converges absolutely
the series diverges
the test is inconclusive.
, then
does not.
Math 1220-002
Camacho
Fall 2012
Midterm #2 Review
Practice Problems
Finite Integration:
Section 7.5:
Section 7.6:
#1 – 40
#1 – 12
Indeterminate Forms:
Section 8.1:
Section 8.2:
#1 – 20
#1 – 44
Improper Integrals:
Section 8.3:
Section 8.4:
#1 – 24
#1 – 32
Infinite Series
Section 9.1:
Section 9.2:
Section 9.3:
Section 9.4:
Section 9.5:
#1 – 20, 21 – 30
#1 – 14, 15 – 24
#1 – 12, 13 – 22, 23 – 26, 27 – 32
#1 – 4, 5 – 10, 11 – 34
# 1 – 6, 7 – 12, 13 – 30