Study Guide for Quiz 3
1. Series with Negative Terms
(a) Absolute Convergence
A series an converges absolutely if the series |an | of absolute values converges. Every
absolutely convergent series converges, but its possible for a series to converge withou
Math 241
Name: _
Homework 1
_
1. One method of defining a sequence +8 8" is to specify the first term, and then give a recursive
formula for the subsequent terms. For example, the equations
+" "
and
+8 #+8" for 8 "
define the sequence e" # % ) "' f. The f
Math 241
Name: _
Homework 2
1. The Koch snowflake is the fractal shape shown in the figures to the right. The
snowflake is constructed from a large equilateral triangle, using the following
process:
i) To each side of the large triangle, attach a smaller
Math 241
Name: _
Homework 4
1. The figure to the right shows a regular hexagon in
three-dimensional space. Find the coordinates of the
point T .
7, 1, 2
P
2, 5, 1
3, 7, 4
0, 0, 1
2. The figure to the right shows a a pyramid in
three-dimensional space.
(a)
5. The Integral Test
Recall that a : -series is a series of the form
"
_
8"
"
8:
where : is some positive constant. As we have previously stated, such a series converges when
: ", and diverges when : ".
Our goal here is to explain this phenomenon. Along t
Math 241
Name: _
Homework 3
1. The figure to the right shows a tiling of the
plane by congruent regular hexagons. Find
the coordinates of the point T .
P
0, 0
1, 0
2. In the figure to the right, a rectangle with a
length of "! units and a width of & units
Math 241
Name: _
Homework 2
1. The Koch snowflake is the fractal shape shown in the figures to the right. The
snowflake is constructed from a large equilateral triangle, using the following
process:
i) To each side of the large triangle, attach a smaller
9. Computing Taylor Series
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 , sin , and cos ), and we
do not have a general formula for findi
4. Convergence and Divergence
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 decimal number:
The sum of this series isn't something simple like or it's just some a
2. Limits at Infinity
To understand sequences and series fully, we will need to have a better understanding of limits at
infinity. We begin with a few examples to motivate our discussion.
EXAMPLE 1
SOLUTION
Find lim
.
Both the numerator and denominator
10. Applications of Taylor Series
These notes discuss three important applications of Taylor series:
1. Using Taylor series to find the sum of a series.
2. Using Taylor series to evaluate limits.
3. Using Taylor polynomials to approximate functions.
Evalu
3. Infinite Series
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 th term is :
We can express an infinite series using summa
7. Series With Negative Terms
Series with a mix of positive and negative terms can behave very differently than series whose
terms are all positive. For a positive series , there are only two possibilities:
1. converges, or
2. (and hence diverges).
As the
1. Sequences
A sequence is an infinite list of numbers written in a definite order.
The members of the list are called terms of the sequence. In the sequence above, the first term
is , the second term is , the third term is , and so forth. There's a notic
6. The Root Test
As we have seen, the key to determining whether a series converges or diverges is to figure out how
quickly its terms go to zero. This is the idea behind both the comparison test and the limit
comparison test.
Unfortunately, there are som
8. Power Series
A power series is a polynomial with infinitely many terms. Here is an example:
Like a polynomial, a power series is a function of . That is, we can substitute in different values
of to get different results. For example,
.
and
.
Thou
Math 241
Name: _
Homework 3
1. We have seen how to use the root test to determine the convergence of exponential series. This test
8 + , but for some series this limit can be hard to compute.
involves the exponential base < lim
8
8_
In these cases, it is