This document provides examples of representing functions as power series. It begins by expressing 1/(1-x) as a power series valid for |x|<1. It then uses this to find power series representations of other functions like 1/(1+x^2), 1/(x+2), and x^3/(x+2) by making substitutions. It also discusses differentiation and integration of power series according to specified rules. Examples find power series for 1/(1-x)^2, ln(1+x), and tan^-1(x) by these operations. All power series are found along with their intervals of convergence.

1. Chapter 11: Sequences, Series, and Power Series
Section 11.9: Representation of Functions as Power Series
Alea Wittig
SUNY Albany

2. Outline
Representations of Functions as Power Series
Differentiation and Integration of Power Series

3. Representations of Functions as Power Series

4. Representations of Functions as Power Series
Recall
f (x) =
1
1 − x
=
∞
X
n=0
xn
for |x| < 1 (1)
▶ We say that
∞
X
n=0
xn
, is the power series representation of
f (x) =
1
1 − x
on the interval (−1, 1).
▶ We can use this power series representation to find the power
series representation of other functions as well.

5. Example 1
1/2
Use (1) to express the function
1
1 + x2
as the sum of a power series and find the interval of convergence.

6. Example 1
2/2
▶ We notice that 1
1+x2 is just the function f (x) = 1
1−x evaluated
at −x2:
f (−x2
) =
1
1 − (−x2)
=
1
1 + x2
▶ Since
1
1 − x
=
∞
X
n=0
xn
for |x| < 1, by plugging in −x2 to both
sides, we have that
1
1 + x2
=
∞
X
n=0
(−x2
)n
for | − x2
| < 1
=
∞
X
n=0
(−1)n
x2n
for |x| < 1
since (−x2
)n
= (−1)n
(x2
)n
= (−1)n
x2n
and
| − x2
| < 1 ⇐⇒ |x2
| < 1 ⇐⇒ |x| < 1

7. Example 2
1/3
Find a power series representation for
1
x + 2

8. Example 2
2/3
▶ Again we use equation (1) by noticing that 1
x+2 is a
transformation of f (x) = 1
1−x :
1
x + 2
=
1
2 + x
=
1
2(1 + (x
2 ))
=
1
2(1 − (−x
2 ))
ie,
1
x + 2
=
1
2
f
−
x
2
.
▶ So we plug in −x
2 to both sides of (1) and get
1
1 − (−x
2 )
=
∞
X
n=0
−
x
2
n
for

11. −
x
2

14. 1
=
∞
X
n=0
(−1)nxn
2n
for |x| 2

15. Example 2
3/3
▶ Now multiplying both sides of the above by 1
2 we get
1
2(1 − (−x
2 ))
=
1
2
∞
X
n=0
(−1)nxn
2n
for |x| 2
=
∞
X
n=0
(−1)nxn
2n+1
for |x| 2
=⇒
1
x + 2
=
∞
X
n=0
(−1)nxn
2n+1
for |x| 2

16. Example 3
1/2
Find a power series representation of
x3
x + 2

17. Example 3
2/2
▶ Now since we found in example 2 that
1
x + 2
=
∞
X
n=0
(−1)nxn
2n+1
for |x| 2
we have that
x3
x + 2
= x3
∞
X
n=0
(−1)nxn
2n+1
for |x| 2
=
∞
X
n=0
(−1)nx3xn
2n+1
for |x| 2
=
∞
X
n=0
(−1)nxn+3
2n+1
for |x| 2

18. Differentiation and Integration of Power Series

19. Differentiation and Integration of Power Series
▶ Theorem: If
P
cn(x − a)n has a radius of convergence R 0,
then the function f defined by
f (x) =
∞
X
n=0
cn(x−a)n
= c0+c1(x−a)+c2(x−a)2
+c3(x−a)3
+. . .
is differentiable on the interval (a − R, a + R) and
(i) f ′
(x) = c1 + 2c2(x − a) + 3c3(x − a)2
+ . . .
=
∞
X
n=1
ncn(x − a)n−1
(ii)
Z
f (x)dx = C + c0(x − a) + c1
(x − a)2
2
+ c2
(x − a)3
3
+ . . .
= C +
∞
X
n=0
cn
(x − a)n+1
n + 1
The radii of convergence in (i) and (ii) are both R.

20. Remark
▶ Note that the intervals of convergence for the derivative and
integral may be different (even though the radii are the same.)

21. Example 4
1/2
Express
1
(1 − x)2
as a power series by differentiating equation (1).