10.4
Power Series and
Taylor’s Theorem
 Recognize power series.
 Find the radii of convergence of power
series.
 Use Taylor’s Theorem to find power series
for functions.
 Use the basic list of power series to find
power series for functions.
1. Power Series
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Power Series
In this section, you will study infinite series that have
variable terms. Specifically, you will study a type of infinite
series that is called a power series.
The index of a power series usually begins with n = 0. In
such cases, assume that
even for x = c.
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Example 1 – Power Series
a. The following power series is centered at 0.
Coefficient
b. The following power series is centered at 1.
c. The following power series is centered at –1.
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2. Radius of Convergence of a
Power Series
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Radius of Convergence of a Power Series
A power series in x can be viewed as a function of x,
where the domain of f is the set of all x for which the power
series converges.
Determining this domain is one of the primary problems
associated with power series. Of course, every power
series converges at its center c because
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which x the power series
conv.
7
Radius of Convergence of a Power Series
8
So, c is always in the domain of f. The domain of a power
series can take three basic forms: a single point, an interval
centered at c, or the entire real number line, as shown in
Figure
(1)
(2)
(3)
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Radius of Convergence of a Power Series
𝐼 maybe = (𝑐 − 𝑅, 𝑐 + 𝑅) , (𝑐 − 𝑅, 𝑐 + 𝑅] ,
[𝑐 − 𝑅, 𝑐 + 𝑅) , or [𝑐 − 𝑅, 𝑐 + 𝑅] .Conv.
div.
div.
c-R
c
c+R
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Radius of Convergence of a Power Series
Determining the convergence or divergence at the
endpoints
can be difficult, and, except for simple cases, the endpoint
question is left open.
c
c
c
c
c
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Example 2 – Finding the Radius of Convergence
Find the radius of convergence of the power series
Solution:
For this power series,
So, you have
So, by the Ratio Test, this series converges for all x, and
the radius of convergence is
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Example 3 – Finding the Radius of Convergence
Find the radius of convergence of the power series
Solution:
For this power series,
. So, you have
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Example 3 – Solution
cont’d
By the Ratio Test, this series will converge as long as
or
So, the radius of convergence is
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Example 3 – Solution
cont’d
Because the series is centered at x = –1, it will converge in
the interval (–3, 1) as shown in Figure .
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3. Taylor and Maclaurin Series
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Taylor and Maclaurin Series
Question 1:
Which functions have power series
representation?
Answer 1: 至少需要在中心點的一鄰域上 具任意階可微 .
我們碰到的函數都可以
Question 2:
How to find such representation?
Answer 2: The problem of finding a power series for a
given function is answered by Taylor’s Theorem.
此課程不去證明了, theorem as below:
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Taylor and Maclaurin Series
This theorem shows how to use derivatives of a function f
to write the power series for f.
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Example 4 – Finding a Maclaurin Series
Find the power series for
centered at 0. What is the radius of convergence of the
series?
Solution:
Begin by finding several derivatives of f and evaluating
each at c = 0.
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Example 4 – Solution
cont’d
From this pattern, you can see that
So, by Taylor’s Theorem,
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Example 4 – Solution
cont’d
From Example 2, you know that the radius of convergence
is
In other words, the series converges for all values of x.
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Example 5 – Finding a Taylor Series
Find the power series for
centered at 1. Then use the result to evaluate
Solution:
Successive differentiation of f(x) produces the pattern
below.
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Example 5 – Solution
cont’d
From this pattern, you can see that
So, by Taylor’s Theorem,
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Example 5 – Solution
To evaluate the series when
sum of a geometric series.
cont’d
use the formula for the
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Taylor and Maclaurin Series
In Example 5, the radius of convergence of the series is
R = 1, and its interval of convergence is (0, 2). (It is
possible to show that the series diverges when x = 0 and
when x = 2.)
Method 2: use geometry series
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Taylor and Maclaurin Series
Compares the graph of
Taylor series for f.
Domain: all x ≠ 0
and the graph of the
Domain: 0 < x < 2
Note that the domains are different. In other words, the
power series in Example 5 represents f only in the interval
(0, 2).
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4. A Basic List of Power Series
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A Basic List of Power Series
The most practical use of Taylor’s Theorem(實際用途) is in
developing power series for a basic list of common
functions.(基本函數冪級數表)
Then, from the basic list, you can determine power series
for other functions by the operations of addition,
subtraction, multiplication, division, differentiation,
integration, and composition with known power series.
we can do so just as we would for a polynomial(逐項操作).
例如
term-by-term differentiation (逐項微分)
and term-by-term integration. (逐項積分) 星期四在加強
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A Basic List of Power Series
今天先建立 課本的五條冪級數
The last series in the list above is called a binomial series.
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其他書
1
 ln(1  x )  
dx   (1  x  x 2  ...) dx
1 x


x2 x3
x n 1
xn
 x    ...  C  
C   C , x 1
2
3
n 0 n  1
n 1 n
Put x = 0 in it and obtain –ln(1 – 0) = C.

x 2 x3
xn
ln(1  x)   x    ...  
2 3
n 1 n
x 1
The radius of convergence is the same as for the
or
original series: R = 1.
ln(1 + x) = x-x2/2+x3/3-x4/4+…. , -1<x≦1
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二項式級數
Find the Maclaurin series for f(x) = (1 + x)k,
where k is any real number
是高中二項式定理
k
k  n
k (k  1)  (k  n  1) n
k  N , (1  x )    x  
x
n!
n 0  n 
n 0
k
k
的推廣
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二項式級數
f ( x )  (1  x ) k
f (0)  1
f '( x )  k (1  x ) k 1
f '(0)  k
f ''( x )  k (k  1)(1  x ) k 2
f ''(0)  k (k  1)
f '''( x )  k (k  1)(k  2)(1  x ) k 3



f '''(0)  k (k  1)( k  2)



f ( n )  k (k  1)  (k  n  1)(1  x ) k n f ( n ) (0)  k (k  1)   (k  n  1)
f ( n ) (0)  k (k  1)  ( k  n  1)


n 0

f ( n ) (0) n
k (k  1)  ( k  n  1) n
x 
x
n!
n!
n 0
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二項式級數
an 1
an


k ( k  1)    ( k  n  1)( k  n ) x n 1
( n  1)!
|k n|
n 1
| x |
1-k/n
| x || x |

n!
k ( k  1)    ( k  n  1) x n
as n  
1+1/n
By Ratio Test,
binomial series conv if |x| < 1 and div if |x| > 1.
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Binomial coefficients
 k  k (k  1)( k  2)  ( k  n  1)  k 
For k  N , 1  n  k ,   
,   : 1
n!
n
 0
 k  k (k  1)( k  2)  ( k  n  1)  k 
For k  R, n  N ,   : 
,  : 1
n!
n
 0
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Example 8 – Using the Basic List of Power Series
Find the power series for each function.
a.
centered at 0
b.
centered at 0
c.
centered at 1
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Example 8(a) – Solution
To find the power series for this function, multiply the series
for ex by 2 and add 1.
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Example 8(b) – Solution
cont’d
To find the power series for
substitute 2x for x
in the series for ex and multiply the result by e.
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Example 8(c) – Solution
To find the power series for
properties of logarithms.
cont’d
centered at 1, use the
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10.4 Homework
上完課會公告 Exercises 10.4 的題目
請交第 7, 11, 19, 25, 29, 31,
33, 37, 41, 45, 49, 53共12 題.
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Test
1. Find a Maclaurin series representation of x3/(x + 2).
2. Find a Maclaurin series representation of
1
3
1+𝑥
.
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solutions

1
( 1)n n
  n 1 x ,  2  x  2
x  2 n 0 2
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