Math 1B
§ 11.9 Representations of Functions as Power Series
Overview: In this section we will represent certain types of functions as power series by manipulating
geometric series or differentiating/integrating such a series. This is useful for integrating functions that
don’t have elementary antiderivatives, for solving differential equations, and for approximating
functions by polynomials.
∞
Recall, the sum of the geometric series is
∑ ar
n= 0
∞
If a = 1 and r = x we have
∑x
n=0
n
=
a
if r < 1
1− r
1
= 1+ x + x 2 + x 3 +... = €
1− x
€
We
€ see that we can represent the function f ( x ) =
€
n
if x < 1
1
with€the power series
1− x
∞
∑x
n
if x < 1
n= 0
A geometric illustration of the above equation is shown below. Since the sum of the series is the limit
€
of the sequence of partial sums, we have
€
€
1
= lim sn (x)
1− x n→∞
where
sn (x) = 1+ x + x 2 + x 3 +... + x n
is the nth partial sum. Notice that as n increases, sn (x) becomes a better approximation to f ( x ) for
−1 < x < 1.
s2(x) = 1 + x + x2
s5(x) = 1 + x + x2 + … + x5
s8(x) = 1 + x + x2 + … + x8
s11(x) = 1 + x + x2 + … + x11
In this section we will represent functions with power series that can be related to
1
1− x
∞
1
= 1+ x + x 2 + x 3 + ... = ∑ x n
1− x
n= 0
€
Stewart – 7e
€
1
Example: Find a power series representation of
1
and find the interval of convergence.
1+ x 3
Example: Find a power series representation of
2x 2
and find the interval of convergence.
1+ x 3
€
Stewart – 7e
2
Example: Find a power series representation of
1
and find the interval of convergence.
5− x
€
Differentiation and Integration of Power Series
∞
n
The sum of a power series is a function f ( x ) = ∑ c n ( x − a) whose domain is the interval of
n= 0
convergence of the series.
We want to differentiate/integrate such functions and the following theorem says we can do so by
differentiating/integrating each€individual term in the series.
∞
Theorem: If the power series
∑ c ( x − a)
n
n
has a radius of convergence R > 0 , then
n= 0
∞
n
2
f ( x ) = ∑ c n ( x − a) = c 0 + c1 ( x − a) + c 2 ( x − a) + ...
n= 0
€
is differentiable (hence
€ continuous) on the interval ( a − R,a + R) and
∞
i) f "( x ) = ∑ nc n ( x − a)
€
ii)
€
€
∫
n−1
2
= c1 + 2c 2 ( x − a) + 3c 3 ( x − a) + ...
€
n=1
n +1
2
3
∞
x − a)
x − a)
x − a)
(
(
(
f ( x ) dx = C + ∑ c n
= C + c 0 ( x − a) + c1
+ c2
+ ...
n +1
2
3
n= 0
Both of these have radius of convergence R.
Stewart – 7e
3
Example: Find a power series representation for f ( x ) =
1
2 and its radius of convergence.
(1− 2x )
€
Stewart – 7e
4
Example: Find a power series representation for f ( x ) = ln(5 − x ) and its radius of convergence.
€
Stewart – 7e
5
Example: Evaluate the indefinite integral as a power series. What is the radius of convergence?
x
∫ 1+ x
Stewart – 7e
3
dx
6