Math 142
Name: _
Homework 1
1. In this problem, you will use the method of partial fractions to evaluate the integral (
B#
"
.B.
%
(a) Find constants E and F for which
B#
"
E
F
.
%
B#
B#
(Hint: You can use any method you like, as long as you find the answ
Series
Lecture Notes
A series is an infinite sum of numbers:
"
"
"
"
#
%
)
"'
The individual numbers are called the terms of the series. In the above series, the first term is
"#, the second term is "%, and so on. The 8th term is "#8 :
"
"
"
"
"
8
#
%
Math 142
Name: _
Homework 4
1. In class, we saw an example of an infinite sum whose value is
finite:
"
"
"
"
"
"
".
#
%
)
"'
$#
'%
(Each term of this sum is half of the preceding term.)
A
B
C
The picture to the right uses this sum to partition a " " squa
Math 142
Name: _
Homework 5
_
1. Consider the infinite series "
8"
8
.
$8
(a) Write out the first ten terms of this series as fractions.
(b) Make a table showing the first ten partial sums. Your answers must be correct to % decimal places.
(c) Based on yo
Math 142
Name: _
Homework 3
1. Consider the region in the first quandrant enclosed by the B-axis and the following parametric curve:
B >$ >
C )># %>$ .
(a) Find the area of this region. (Feel free to use a calculator to compute the integral.)
(b) Suppose
Sequences
Lecture Notes for Section 8.1
A sequence is an infinite list of numbers written in a definite order:
#
%
)
"'
$#
The numbers in the list are called the terms of the sequence. In the sequence above, the first
term is #, the second term is %, the
Math 142
Name: _
Homework 2
1. The figure to the right shows two circles, one with radius &
and one with radius $; these circles intersect at the points
a$ ' and a$ '.
The shaded, crescent-shaped region shown in the figure is
called a lune.
(a) Find the e
Power Series
Lecture Notes
A power series is a polynomial with infinitely many terms. Here is an example:
0 aB b " B B # B $
Like a polynomial, a power series is a function of B. That is, we can substitute in different
values of B to get different result
Convergence and Divergence
Lecture Notes
It is not always possible to determine the sum of a series exactly. For one thing, it is common for
the sum to be a relatively arbitrary irrational number:
"
_
8"
"
"
"
"
" # $ % "#*"#)'
8
8
#
$
%
The sum of this
Convergence of Power Series
Lecture Notes
Consider a power series, say
0 a B b " B B # B $ B % .
Does this series converge? This is a question that we have been ignoring, but it is time to face it.
Whether or not this power series converges depends on the
Computing Taylor Series
Lecture Notes
As we have seen, many different functions can be expressed as power series. However, we
do not yet have an explanation for some of our series (e.g. the series for /B , sin B, and cos B), and
we do not have a general f
The Root Test
Lecture Notes
So far, we have learned how to use the limit comparison test to determine whether a series
converges or diverges. The idea of the limit comparison test is that a series will converge as long
as its terms go to zero quickly enou
Math 142
Name: _
Homework 1
1. In this problem, you will use the method of partial fractions to evaluate the integral (
B#
"
.B.
%
(a) Find constants E and F for which
B#
"
E
F
.
%
B#
B#
(Hint: You can use any method you like, as long as you find the answ