March Wednesday, 7 Lecture 26 : Expressing functions as a power series:
Derivatives and Integrals of power series. (Refers to Sections 8.6 and 8.7 )
After having practiced the problems associated to the concepts of this lecture the student should be
able to: Express a function which is a relative of the geometric function 1/(1 x) as a power series.
26.1 Questions In previous examples we saw that the geometric series j=0 to x j
converges to the function 1/(1 x) on the interval (1, 1). We can then also say that the
function 1/(1 x) has a power series representation. We wonder if there are other
functions which have a power series representation. In this lecture we will see that there are
many. In particular those who, in some way, are related to the function 1/(1 x). Look
carefully at the following examples to see this:
Observe that the above equalities only hold for certain values of x. The theorem below shows
that derivatives and the integrals of functions which are relatives of 1/(1 x) are functions
for which we can also obtain power series representations.
Below we will work with functions which are relatives of the geometric function 1/(1 x)
by differentiation and integration.
26.1.1 Example Find a power series representation of the function 1/(1 x)2.
Solution: We will make use of the form
() =
(), for any
Note that in the chain of equalities below we must restrict x to x < 1.
26.1.2 Example Find a power series for the function f(x) = ln(1 + x) and give its
interval convergence.
Solution:
Recall of that
() () =
()
To find the interval of convergence we must test convergence at x = 1 and x = 1.
-
-
The function ln(1 + x) is not defined at x = 1 so we need only test for
convergence at x = 1.
When x = 1, this series is the alternating harmonic series which is known to
converge.
So the series converges to ln(1 + x) on the interval (1, 1]
26.2 Example Express the function ln x as a power series. (Note that (d/dx) ln x = 1 /x.)
26.3 Example Show that f(x) = arctan x can be expressed as the power series
Verify that its radius of convergence is R = 1. We only outline the solution by setting up the integral:
26.4 Example Express the function f(x) = 1/(x 2) as a power series centered at 3 and
state the radius of convergence as well as the interval of convergence.
Radius of convergence R = 1, and interval of convergence is (2, 4).
25.5 Example Express the function
as a power series centered at 4 and state the radius of convergence as well as the interval
of convergence. Note: The function 2x2 5x + 3 factors to (x 1)(2x 3) and see that
f (x) the expression can be broken up in partial fractions as follows:
Solution:
26.6 Example Express the function
as a power series centered at 0 and state the radius of convergence as well as the interval
of convergence.
26.7 Example Find a power series representation for the function
centered at 0 and state the radius of convergence as well as the interval of convergence.
Solution: