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8.6. POWER SERIES
Power Series
As Preview Activity 8.6 shows, it can be useful to treat an unknown function as if it has
a Taylor series, and then determine the coefficients from other information. In other
words, we define a function as an infinite series of powers of x and then determine the
coefficients based on something besides a formula for the function. This method of using
series illustrated in Preview Activity 8.6 to solve differential equations is a powerful and
important one that allows us to approximate solutions to many different types of differential
equations even if we cannot explicitly solve them. This approach is different from defining
a Taylor series based on a given function, and these functions we define with arbitrary
coefficients are given a special name.
Definition 8.9. A power series centered at x = a is a function of the form
∞
k=0
c k (x − a)
k
(8.26)
where {c k } is a sequence of real numbers and x is an independent variable.
We can substitute different values for x and test whether the resulting series converges
or diverges. Thus, a power series defines a function f whose domain is the set of x values
for which the power series converges. We therefore write
f (x) =
∞
k=0
c k (x − a)
k .
It turns out that 7 , on its interval of convergence, a power series is the Taylor series of
the function that is the sum of the power series, so all of the techniques we developed in
the previous section can be applied to power series as well.
Example 8.6. Consider the power series defined by
f (x) =
∞
k=0
x k
2 k .
What are f (1) and f
3
2
? Find a general formula for f (x) and determine the values for
which this power series converges.
7 See Exercise 2 in this section.