Third Quarter Calculus for Engineering and other Hard Sciences
MATH 107

Spring 2003
Power Series
Definition of a Power Series
Definition of a Power Series
Let f(x) be the function represented by the series
Then f(x) is called a power series function.
More generally, if f(x) is represented by the series
Then we call f(x) a power series ce
Third Quarter Calculus for Engineering and other Hard Sciences
MATH 107

Spring 2003
Name
MATH 107 PRACTICE MIDTERM III
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Show your work.
PROBLEM 1 Please answer the following true or false. If false, explain why or provide
Third Quarter Calculus for Engineering and other Hard Sciences
MATH 107

Spring 2003
Sequences
Definition of a Sequence
A sequence is a list of numbers, or more formally, a function f(n) from the natural numbers to the
real numbers.
We write
an
to mean the nth term of the sequence.
Example:
If
1
an =
n+2
then we have
a1 = 1/3,
a2 = 1/4,
e
Third Quarter Calculus for Engineering and other Hard Sciences
MATH 107

Spring 2003
MATH 107 PRACTICE MIDTERM 1
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Show your work.
PROBLEM 1 Please answer the following true or false. If false, explain why or provide a
count
Third Quarter Calculus for Engineering and other Hard Sciences
MATH 107

Spring 2003
Name
MATH 107 PRACTICE FINAL
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Show your work.
PROBLEM 1 Please answer the following true or false. If false, explain why or provide a
coun
Third Quarter Calculus for Engineering and other Hard Sciences
MATH 107

Spring 2003
Name
MATH 107 PRACTICE MIDTERM II
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Show your work.
PROBLEM 1 Please answer the following true or false. If false, explain why or provide a
Third Quarter Calculus for Engineering and other Hard Sciences
MATH 107

Spring 2003
Series
Definition of a Series
Let an be a sequence then we define the nth partial sum of an as
sn = a1 + a2 + . + an
In other words, we define sn by adding up the first n terms of an. We define the series as the limit
of the sn that is
S = an = a1 + a2 +
Third Quarter Calculus for Engineering and other Hard Sciences
MATH 107

Spring 2003
Creating Power Series From Functions
The Geometric Power Series
Recall that
Substituting x for r, we have
We write
Milking the Geometric Power Series
By using substitution, we can obtain power series expansions from the geometric series.
Example 1
Substit
Third Quarter Calculus for Engineering and other Hard Sciences
MATH 107

Spring 2003
Taylor Polynomials
Review of the Tangent Line
Recall that if f(x) is a function, then f '(a) is the slope of the tangent line at x = a. Hence
y  f(a) = f '(a) (x  a)
or
P1(x) = y = f(a) + f '(a) (x  a)
is the equation of the tangent line. We can say th
Third Quarter Calculus for Engineering and other Hard Sciences
MATH 107

Spring 2003
Comparison Tests
The Direct Comparison Test
If a series "looks like" a geometric series or a Pseries (or some other known series) we can use
the test below to determine convergence or divergence.
Theorem: The Direct Comparison Test
Let
0 < an < bn
for al
Third Quarter Calculus for Engineering and other Hard Sciences
MATH 107

Spring 2003
Alternating Series
The Alternating Series Test
Suppose that a weight from a spring is released. Let a1 be the distance that the spring drops on
the first bounce. Let a2 be the amount the weight travels up the first time. Let a3 be the amount
the weight tr
Third Quarter Calculus for Engineering and other Hard Sciences
MATH 107

Spring 2003
Taylor Series
Taylor Series
Recall that the Taylor polynomial of degree n for a differentiable function f(x) centered
at x = c is
If we let n approach infinity, we arrive at the Taylor Series for f(x) centered at x = c.
Definition
The Taylor Series for f(
Third Quarter Calculus for Engineering and other Hard Sciences
MATH 107

Spring 2003
Root and Ratio Test
The Ratio Test
Theorem: The Ratio Test
Let
1.
be a series then
If
then the series converges absolutely.
2.
If
then the series diverges.
3.
If
then try another test.
Proof: Suppose that
then for the tail,
an+1 < R an
an+2 < R an+
Third Quarter Calculus for Engineering and other Hard Sciences
MATH 107

Spring 2003
Integral Test and pSeries
The Integral Test
Consider a series an such that
an > 0
and
an > an+1
We can plot the points (n,an) on a graph and construct rectangles whose bases are of length 1 and
whose heights are of length an. If we can find a continuous