Toán 3

Lesson 3: Power Series (11.8-11.10)

(Chuỗi lũy thừa)
Nguyen Chanh Tu, DUT

(From Maple worksheet)

11.7. Team project part 1

All teams work out applications of serries, use Sec. 11.7 as the first part of the team final project.
O

11.8. Power Series (Chuỗi lũy thừa)

Definitions and examples
Definitions (R) (Chuỗi lũy thừa)
1) A power series is a series of the form
N

> c x = c Cc x Cc x Cc x C....
n=0
n
n
0 1 2
2
3
3

where x is a variable and the cn’s are constants called the coefficients of the series.
For each fixed x, the above series is a series of constants that we can test for convergence or
divergence.

A power series may converge for some values of x and diverge for other values of x. The sum
of the series is a function
f x = c0 Cc1 x Cc2 x2 Cc3 x3 C....Ccnxn C...
whose domain is the set of all values of x for which the series converges.
Miền xác định của chuỗi lũy thừa.
2) (Chuỗi lũy thừa tổng quát) More generally, a series of the form
N

>c
n= 0
n x Ka n
= c0 Cc1 x Ka Cc2 x Ka 2 Cc3 x Ka 3 C....

is called a power series in x Ka or a power series centered at a or a power series about a.

(....gọi là chuỗi lũy thừa của (x-a) hay chuỗi lũy thừa tâm a)
Examples

Ex.1
For instance, if we take cn = c for all n , the power series becomes the geometric series
N

> c x = c Cc x Ccx Cc x C....= ?

n= 0
n 2 3

which converges when x ! 1 and diverges otherwise.

 N

>c x
n=0
n
=
c
1 Kx
= f x when x ! 1. It means that the domain of f x is x ! 1.

O
Ex. 2
N
For what values of x is the series > n!x convergent?
n=0
n

N
It means that you need to find the domain of the function: f x = > n!x .
n= 0
n

Solution
O a d n/n!$xn
a := n/n! xn (2.1.2.2.1.1)
a n C1
O simplify ;
a n
x n C1 (2.1.2.2.1.2)
Then, the given series converges only when x = 0.
O
Ex. 3
N n
For what value of x is the series >
n=1
x K2
n
convergent?

Solution
Ex. 4. Bessel function
Find the domain of the Bessel function of order 0 defined by
N
K1 nx2n
J0 = >
n= 0 2
2n
n! 2
Solution

Radius and interval of convergence (Bán kính hội tụ-khoảng hội tụ)
Definition and Theorem

Definition: The number R in case (iii) is called the radius of convergence of the power
series. By convention, the radius of convergence in case (i) is R = 0 and in case (ii) is
R =N.
 The interval of convergence of a power series is the interval that consists of all values of
x for which the series converges.
O N
Examples
In general, the Ratio Test (or sometimes the Root Test) should be used to determine the radius
of convergence . The Ratio and Root Tests always fail when x is an endpoint of the interval of
convergence, so the endpoints must be checked with some other test.

Ex. 1
Find the radius and interval of convergence of the series
N
K2 nxn
>
n=0 n C3
Sol
Ex. 2
Find the radius and interval of convergence of the series
N n
>
n=0
n x C3
5n C 1
Sol

11.9. Representations of Functions as Power Series

Definition (R)
N
• Recall that the sum of the series > c x is a function
n= 0
n
n

f x = c0 Cc1 x Cc2 x2 Cc3 x3 C....Ccnxn C...

whose domain is the set of all x for which the series converges.

• Now we are given a function f x , the question is if we can express the function f as the sum of
a power series.

c
For example, let f x = then we know that
1 Kx
 N
f x =
c
1 Kx
= c Ccx Ccx2 Ccx3 Ccx4 Ccx5 C...= >
n= 0
c xn, x ! 1.

Note that, it is well-defined only when x ! 1.

That is a representation of f x as the sum of a power series.

O

Examples
Express each of the following functions as the sum of a power series and find the interval of
convergence.
2 1
a) f x = b) g x =
1 Cx
2 3 Cx
x2
c) h x =
x C3
O int 2 / 1 Cx^2 , x
2 arctan x (3.2.1)
Sol a)
sol b)

sol c)

Differentiation and Integration of Power Series (R)

Examples
Exam.1

3
O restart; f d x / 2
x K5
3
f := x/ (3.3.1.1.1)
x K5 2
We want to find a power series representation of f x . We see that.
O Int f x , x : % = value % ; F d int f x , x ;
3 3
dx = K
x K5
2 x K5
3
F := K (3.3.1.1.2)
x K5
O
The function F can be expressed as the sum of a geometric series.
O series F, x = 0, 5 ;
3 3 3 3 3 3
C xC x2 C x C x4 CO x5 (3.3.1.1.3)
5 25 125 625 3125
Take the differential of the terms of the series.
O diff %, x ; g d convert %, polynom ;
3 6 9 2 12 3
C xC x C x CO x4
25 125 625 3125
(3.3.1.1.4)
 3 6 9 2 12 3
C
g := xC x C x (3.3.1.1.4)
25 125 625 3125
Now look at the graphes of f and g.
O plot f x , g , x =K3 ..3, color = red, blue , thickness = 3, 2 ;

0.7

0.6

0.5

0.4

0.3

0.2

0.1

K3 K2 K1 0 1 2 3
x
O
The above example tell us that
N
3
x K5 2
= > dxd
k= 0 5
3
kC1
xk

or

d 3 d 3 d 3 d 3 2 d 3 3
= C x C x C x C...
dx 5 Kx dx 5 dx 25 dx 125 dx 625
O
Theorem (R)
Note: We can write the above equalities as:
N N
•
d
dx >c
n= 0
n x Ka n
=
n=0
> d
c x Ka
dx n
n

N N
• >c
n=0
n x Ka n
dx=C C >
n=0
cn x Ka ndx

O
Applications
Ex.1
1 1
Express as a power series by differentiating the power series of . What is
1 Kx
2 1 Kx
the radius and interval of convergence?
Sol
Ex. 2
Find a power series representation for ln 3 C x and its radius and interval of convergence.

Sol
O
Ex. 3
1
a) Evaluate 7
dx as a power series.
1 Cx
 1
O restart; int 7
,x
1 Cx
1
7 >
_R = RootOf _Z6 C _Z5 C _Z4 C _Z3 C _Z2 C _Z C 1
_R ln x C_R C
1
7
ln x C1 (3.3.3.3.1)

O
0.5
1
b) Use part (a) to approximate 7
dx correct to within 10K7.
1 Cx
0
Hint a)
1
Can you express the function as the sum of a power series ?
1 Cx7
Solution
O

Hint b)

Sol
O

11.10. Taylor Series and Maclaurin series

Theorem about Representations of Functions as Power Series

Taylor Series and Maclaurin series

Taylor polynomials and approximations

HW
Sec. 11.7
1) From 1-20: at least 5
2) From 21-38 at least 5
Sec. 11.8
I. 1,2, 29
II 3-28 at least 8
III. Teamwork
1. 29-33
2. 34-40 at least 6
Sec. 11.9
I. 1,2,
II
3-10 at least 4
15-18: al. 2
19-22 al. 2
23-26 al. 2
 27-30 al. 2
III. Teamwork
31-40 al. 6
O
O