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8.6. POWER SERIES
8.6 Power Series
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a power series?
• What are some important uses of power series?
• What is the connection between power series and Taylor series?
Introduction
We have noted at several points in our work with Taylor polynomials and Taylor series that
polynomial functions are the simplest possible functions in mathematics, in part because
they essentially only require addition and multiplication to evaluate. Moreover, from the
point of view of calculus, polynomials are especially nice: we can easily differentiate or
integrate any polynomial. In light of our work in Section 8.5, we now know that several
important non-polynomials have polynomial-like expansions. For example, for any real
number x,
e
x = 1 + x +
x 2
2!
+
x 3
3!
+ · · · +
x n
n!
+ · · · .
As we continue our study of infinite series, there are two settings where other series like
the one for e x arise: one is where we are simply given an expression like
1 + 2x + 3x
2 + 4x
3 + · · ·
and we seek the values of x for which the expression makes sense, while another is where
we are trying to find an unknown function f , and we think about the possibility that the
function has expression
f (x) = a 0 + a 1 x + a 2 x
2 + · · · + a k x
k + · · · ,
and we try to determine the values of the constants a 0 , a 1 , . . .. The latter situation is
explored in Preview Activity 8.6.
Preview Activity 8.6. In Chapter 7, we learned some of the many important applications
of differential equations, and learned some approaches to solve or analyze them. Here, we
consider an important approach that will allow us to solve a wider variety of differential
equations.
Let’s consider the familiar differential equation from exponential population growth
8.6. POWER SERIES
8.6 Power Series
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a power series?
• What are some important uses of power series?
• What is the connection between power series and Taylor series?
Introduction
We have noted at several points in our work with Taylor polynomials and Taylor series that
polynomial functions are the simplest possible functions in mathematics, in part because
they essentially only require addition and multiplication to evaluate. Moreover, from the
point of view of calculus, polynomials are especially nice: we can easily differentiate or
integrate any polynomial. In light of our work in Section 8.5, we now know that several
important non-polynomials have polynomial-like expansions. For example, for any real
number x,
e
x = 1 + x +
x 2
2!
+
x 3
3!
+ · · · +
x n
n!
+ · · · .
As we continue our study of infinite series, there are two settings where other series like
the one for e x arise: one is where we are simply given an expression like
1 + 2x + 3x
2 + 4x
3 + · · ·
and we seek the values of x for which the expression makes sense, while another is where
we are trying to find an unknown function f , and we think about the possibility that the
function has expression
f (x) = a 0 + a 1 x + a 2 x
2 + · · · + a k x
k + · · · ,
and we try to determine the values of the constants a 0 , a 1 , . . .. The latter situation is
explored in Preview Activity 8.6.
Preview Activity 8.6. In Chapter 7, we learned some of the many important applications
of differential equations, and learned some approaches to solve or analyze them. Here, we
consider an important approach that will allow us to solve a wider variety of differential
equations.
Let’s consider the familiar differential equation from exponential population growth